|Jane Austen by her sister Cassandra, ca. 1810|
A forbidden marriage
Cecilia is an heiress, but a condition of her inheritance is that, if she marries, her husband must take her name. She meets and falls in love with Mortimer Delvile, the only son of a proud and ancient family. His parents forbid their union, leaving Cecilia miserable:
...her fate was finally determined, and its determination was not more unhappy than humiliating; she was openly rejected by the family whose alliance she was known to wish; she was compelled to refuse the man of her choice, though satisfied his affections were her own. A misery so peculiar she found hard to support, and almost bursting with conflicting passions, her heart alternately swelled from offended pride, and sank from disappointed tenderness. In December 1795, when she was twenty, Jane Austen met Tom Lefroy. He had recently graduated from Trinity College, Dublin, and was on his way to study law in London when he visited his aunt and uncle in Ashe, a small Hampshire village about 50 miles southwest of London. Ashe is also about two miles from Steventon, where Jane was born and had lived her entire life; the Ashe Lefroys and the Austens were well acquainted.
It was the Christmas season, and Jane and Tom paid courtesy calls to each other's houses, and danced together at several balls. On Saturday 9 January, 1796, the morning after a ball, Jane wrote in a letter to her sister Cassandra,
...I am almost afraid to tell you how my Irish friend and I behaved. Imagine to yourself everything most profligate and shocking in the way of dancing and sitting down together. I can expose myself however, only once more, because he leaves the country soon after next Friday, on which day we are to have a dance at Ashe after all. He is a very gentlemanlike, good-looking, pleasant young man, I assure you...[H]e has but one fault, which, I trust, time will entirely remove—it is that his morning coat is a great deal too light. In a letter to Cassandra dated the following Thursday, Jane wrote of the dance at Ashe to be given the next night:
I look forward with great impatience to it, as I rather expect to receive an offer from my friend in the course of the evening. I shall refuse him, however, unless he promises to give away his white coat.Tom Lefroy was dependent on a great-uncle for support, and so was not free to marry a woman with no fortune such as Jane Austen. Stories persisted within the Lefroy family for decades that Tom had been sent away by his aunt, out of concern for Jane, to prevent this flirtation from turning more serious.
...Tell Mary that I make over Mr. Heartley and all his estate to her for her sole use and benefit in the future, and not only him, but all my other admirers into the bargain wherever she can find them, even the kiss which C. Powlett wanted to give me, as I mean to confine myself in future to Mr. Tom Lefroy, for whom I don't care sixpence.
Friday.—At length the day is come on which I am to flirt my last with Tom Lefroy, and when you receive this it will be over.—My tears flow as I write, at the melancholy idea...There is a report that Tom is to be married to a Lichfield lass. 
There is strong evidence that Tom indeed fell in love with Jane. In 1799 he married Mary Paul; their first daughter, born in 1802, was named Jane Christmas Lefroy. While Jane was also the name of Mary Paul's mother, the addition of "Christmas" is almost certainly an allusion to the season of Tom's first meeting with Jane Austen, and suggests that Jane Christmas is Austen's namesake.
And in 1870 Tom's nephew Thomas Edward Preston Lefroy wrote to Jane's nephew James Edward Austen-Leigh, who had just published a memoir of Jane and was preparing a second edition, "My late venerable uncle...said in so many words that he was in love with her, although he qualified his confession by saying it was a boyish love. As this occurred in a friendly & private conversation, I feel some doubt whether I ought to make it public."  In the second edition of the memoir, Austen-Leigh wrote only that Lefroy would "remember and speak, as he sometimes did, of his former companion, as one to be much admired, and not easily forgotten by those who had ever known her." 
There has been speculation about the extent to which Pride and Prejudice's Mr. Darcy (or Sense and Sensibility's Mr. Willoughby) may be modelled on Tom Lefroy. However, I'd like to suggest that he is (also?) represented in another character: Darcy's cousin Colonel Fitzwilliam. He and Elizabeth engage in mild flirtation while she is visiting Hunsford Parsonage. In conversation with her one day he tells her that he will be leaving soon, and then warns her (and himself?) that he is not free to bestow his heart:
"Younger sons cannot marry where they like."
"Unless where they like women of fortune, which I think they very often do."
"Our habits of expense make us too dependent, and there are not many in my rank of life who can afford to marry without some attention to money."
"Is this," thought Elizabeth, "meant for me?" and she coloured at the idea; but, recovering herself, said in a lively tone, "And pray, what is the usual price of an earl's younger son? Unless the elder brother is very sickly, I suppose you would not ask above fifty thousand pounds."
He answered her in the same style, and the subject dropped. 
|Jane Austen by her sister Cassandra, ca. 1804|
A change of heart
In Cecilia, Mortimer Delvile finally persuades the heroine, despite his family's opposition, to marry him—and to have the wedding in London in just a few days' time. However, no sooner does she agree to his plan than she regrets it, and spends a sleepless night:
Left now to herself, sensations unfelt before filled the heart of Cecilia. All that had passed for a while appeared a dream; her ideas were indistinct, her memory was confused, her faculties seemed all out of order, and she had but an imperfect consciousness either of the transaction in which she had just been engaged, or of the promise she had bound herself to fulfil; even truth from imagination she scarcely could separate; all was darkness and doubt, inquietude and disorder!And yet, she does decide to refuse him. As the quote above suggests, her vacillation is not capricious, but emotionally agonizing.
But when at length her recollection more clearly returned, and her situation appeared to her such as it really was, divested alike of false terrors or delusive expectations, she found herself still farther removed from tranquillity.
Hitherto, though no stranger to sorrow...she had yet invariably possessed the consolation of self-approving reflections; but the step she was now about to take, all her principles opposed; ...scarce was Delvile out of sight, before she regretted her consent to it.
...Yet to disappoint Delvile so late, by forfeiting a promise so positively accorded; to trifle with a man who to her had been uniformly candid, to waver when her word was engaged, and retract when he thought himself secure,—honour, justice and shame told her the time was now past. 
Jane Austen also spent 24 hours in an emotional quandary over a proposal. In late November 1802 Jane and her sister Cassandra made an extended visit to their longtime friends Catherine and Alethea Bigg at their family home Manydown, about five miles from Steventon (although at this time Jane and Cassandra were living in Bath with their parents; her brother James and his family were living in Steventon). A week into the visit, Harris Bigg-Wither, Catherine and Alethea's brother, proposed to Jane. James Austen's daughter Caroline later wrote:
Mr. Wither was very plain in person—awkward, & even uncouth in manner—nothing but his size to recommend him—he was a fine big man—but one need not look about for secret reason to account for a young lady's not loving him—a great many would have taken him without love—... Harris made his offer on the evening of 2 December, and Jane accepted him. But the next morning she withdrew her consent, and she and Cassandra abruptly left Manydown. Caroline continued,
I conjecture that the advantages he could offer, & her gratitude for his love, & her long friendship with his family, induced my Aunt to decide that she would marry him when he should ask her—but that having accepted him she found she was miserable & that the place & fortune which would certainly be his, could not alter the man...To be sure she should not have said yes—over night—but I have always respected her for the courage in cancelling that yes—the next morning. Several decades afterwards, one of Jane's nieces, Catherine Hubback, read the letters Jane later wrote to Cassandra referring to this incident (Cassandra eventually destroyed these letters and many others written by Jane); Catherine wrote, "I gathered from the letters that it was in a momentary fit of self-delusion that Aunt Jane accepted Mr Wither's proposal, and that when it was all settled eventually, and the negative decisively given she was much relieved. I think the affair vexed her a great deal..." 
Cecilia doesn't want to act dishonorably in marrying the man she loves; Jane Austen didn't want to act dishonorably in marrying a man she didn't love. In Pride and Prejudice, on learning of her sister Elizabeth's engagement to a man she is believed to disdain, Jane Bennet says, "'And do you really love him quite well enough? Oh, Lizzy! do anything rather than marry without affection.'"  And in an example of life imitating art imitating life, when Jane's niece Fanny sought her advice about an admirer, Jane wrote:
I shall turn round and entreat you not to commit yourself farther, and not to think of accepting him unless you really do like him. Anything is to be preferred or endured rather than marrying without affection; and if his deficiences of manner, &c. &c., strike you more than all his good qualities, if you continue to think strongly of them, give him up at once. 
An objection, and a response
I've proposed that one reason Jane Austen esteemed Fanny Burney's Cecilia so highly is that she recognized her own emotional quandaries in those of its heroine—in particular, her feelings when faced with a forbidden marriage and a change of heart over a marriage proposal. However, one possible objection to this idea is that Northanger Abbey, the novel in which Austen's narrator specifically praises Cecilia, was first written in 1798-99. While this is after the flirtation with Tom Lefroy, it occurs before Harris Bigg-Wither's proposal of marriage.
However, it's known that in 1803, just a few months after her acceptance and next-day refusal of Bigg-Wither's proposal, Jane Austen revised the manuscript of Northanger Abbey (then entitled Susan) before submitting it anonymously to a publisher. And in 1816, Jane's brother Henry was able to buy the still-anonymous manuscript back for the same £10 that had been paid for it by the publisher; Jane revised the manuscript again before her untimely death in July 1817.
If her own experiences inspired her narrator's praise of Fanny Burney's novels and their insight into the human heart, that praise might have been added during either the 1803 or the 1817 revision. As additional evidence, the passage in its final form must have post-dated the novel's initial composition, because Maria Edgeworth's Belinda (published in 1801) is included in the narrator's praise. Unfortunately only a fragment of the Susan manuscript survives, and so this is a question that, like many relating to Austen's life and work, may never be definitively resolved.
For other posts on Jane Austen and Fanny Burney, please see:
1. Jane Austen, Northanger Abbey, Ch. V.
2. Fanny Burney, Cecilia, Book VII, Ch. VIII.
3. Jane Austen's Letters to her sister Cassandra and others, R.W. Chapman, ed. Clarendon Press, Oxford, 1932. Vol. I, pp. 1-3. (JAL)
4. JAL, Vol. I, pp. 5-6.
5. Deirdre Le Faye, Jane Austen: A family record, 2nd edition. Cambridge University Press, 2004. p. 278. (JAFR) This book is an essential resource for documents relating to Jane Austen's life.
6. James Austen-Leigh, A memoir of Jane Austen by her nephew. Folio Society, London, 1989, p. 50.
7. Jane Austen, Pride and Prejudice, Ch. 33.
8. Cecilia, Book VII, Ch. III.
9. JAFR, pp. 137-138.
10. JAFR, p. 138.
11. JAFR, p. 138.
12. Pride and Prejudice, Ch. 59.
13. JAL, Vol. II, p. 410.