In A Room of One's Own, Virginia Woolf speculates about what the life of Shakespeare's sister might have been like had she existed. If she had been equally as talented, and equally "as adventurous, as imaginative, as agog to see the world" as her famous brother, she would have been prevented—thanks to the constraints placed on Elizabethan women—from making use of her gifts.
But some great artists really did have sisters. When Wolfgang Mozart was a child prodigy he toured Europe giving concerts with his older sister Maria Anna, known in the family as Nannerl. Nannerl was clearly very talented; her harpsichord playing was described by observers as "masterly" and "brilliant." In a letter published in the Augsburger Intelligenz-Zettel of May 19, 1763, an anonymous reviewer wrote, "Just imagine a girl 11 years of age who can perform on the harpsichord or fortepiano the most difficult sonatas and concertos by the greatest masters, most accurately, readily and with an almost incredible ease, in the very best of taste." 
Nannerl also wrote music; in a letter from 1770 Wolfgang praises one of her songs, writing "I am amazed to find out how well you can compose. In a word, the song is beautiful. Try this more often."  Unfortunately, none of her compositions have survived.
But when Nannerl was 18, her father Leopold began to focus all of his attentions (and the family resources) on promoting Wolfgang's career. Nannerl and her mother were left behind in Salzburg as Wolfgang and Leopold toured Italy and travelled to Vienna. Wolfgang wrote many piano pieces that were clearly intended for performance by Nannerl, including his first piano duet (K381, from 1772). The conductor and musicologist Jane Glover writes of this piece, "It is significant that there is absolute parity between the Primo and Secondo parts; as a player, Nannerl was entirely Wolfgang's equal."  But while she played with her brother in public occasionally over the next decade, her life as a performer began to draw to a close, and ended entirely with her marriage in 1783.
Fanny Mendelssohn was Felix Mendelssohn's older sister, and received musical instruction from the same tutors as Felix. Like Nannerl, Fanny was a pianist and composer. But because she came from an upper-class family performances for paying customers were considered unseemly, and so Fanny played only in the family home and for invited guests. When she was a young woman her father wrote her, "Music will perhaps become his [Felix's] profession, while for you it can and must be only an ornament, never the root of your being and action....You must...prepare more earnestly and eagerly for your real calling, the only calling of a young woman—I mean the state of a housewife." When their mother urged Felix to help Fanny publish her music, he wrote her that to do this "is contrary to my views and to my convictions." 
Fanny continued to compose, however, even after her marriage to the painter William Hensel (who fully supported her musical activities); ultimately she produced more than 400 works. It may have been contrary to Felix's convictions to publish Fanny's works under her own name; however, he included six of her songs in his song collections Opus 8 and Opus 9. Finally in 1846 Fanny began to publish her songs and piano pieces under her own name; unfortunately she died of a stroke the following year, and after a few more works were published at William Hensel's request, nothing more appeared until 1987.
Today, however, Fanny Mendelssohn Hensel is recognized as one of the most significant 19th-century composers, and a definitive biography of Fanny by R. Larry Todd has recently been published (Fanny Hensel: The Other Mendelssohn, Oxford, 2009). In her post "Celebrating Fanny Hensel," Jessica Duchen has conducted a fascinating interview with Todd and included a performance of one of Fanny's duets, "Aus meinen Tränen sprießen viel blühende Blumen hervor" (From my tears spring many blooming flowers), sung by Barbara Bonney and Angelika Kirchschlager with pianist Malcolm Martineau:
The words in German and English can be found on WikiBooks.
Many thanks to acatalano2641 for the video and to Jessica for including it in her wonderful post on Fanny, which inspired mine. There is also a website devoted to Fanny which includes links to CDs, books, sheet music, and concert performances of her work (mainly in Germany).
Update 9 March 2017: Fanny Mendelssohn Hensel's "Easter Sonata" for piano was recently rediscovered by musicologist Angela Mace Christian after being misattributed to Felix for decades. Fanny's great-great-great-granddaughter, Sheila Hayman, recounts the discovery in The Guardian UK of 8 March 2017: "A Fanny Mendelssohn masterpiece finally gets its due."
 Quoted in Otto Eric Deutsch's Mozart: A Documentary Biography, Stanford University Press, 1966, pp. 17-26.
 From letter 102a in Emily Anderson's Letters of Mozart and His Family, Vol. 1, MacMillan, 1938, p. 219.
 From Jane Glover's Mozart's Women: His Family, His Friends, His Music, Harper Collins, 2005, pp. 43-44.
 Quoted in Carol Neuls-Bates's Women in Music: An Anthology of Source Readings from the Middle Ages to the Present. Northeastern University Press, 1996, pp. 144-148.