In Act 3, Cathleen prods a delirious Mary, asking questions about her past. In a long and beautiful monologue, Mary describes her hands by flashing back to her childhood in the convent:
They were a musician's hands. I used to love the piano. I worked so hard at my music in the convent—if you can call it work when you do something you love. Mother Elizabeth and my music teacher both said I had more talent than any student they remembered. My father paid for special lessons. He spoiled me. He would do anything I asked. He would have sent me to Europe to study after I graduated from the Convent. I might have gone—if I hadn't fallen in love with Mr. Tyrone. Or I might have become a nun. I had two dreams. To be a nun, that was the more beautiful one. To become a concert pianist, that was the other.
The lives of the Tyrone family before the events of the play are, for the most part, a mystery to the reader. The family's past is shrouded in their anxieties and only communicated in the play in vague implications. But this monologue from Mary is an exception, providing some of the clearest backstory that O'Neill gives for any of his characters. Audiences see clearly that Mary fondly remembers the convent as the happiest time of her life. She had talent on the piano and was loved deeply for it by people she respected, and she felt connected to a strong religious faith. But then, Mary left the convent, after she had "fallen in love with Mr. Tyrone." She abandoned both of her dreams to be a nun or concert pianist. This backstory shows the internal turmoil that causes Mary's deep anxieties. Mary feels so deeply torn by her love for Tyrone (and it is unclear if she still does love him when the play takes place) and the dreams she lost by marrying him.
This backstory informs every other time Mary has discussed her hands in the play. Her hands are the nominal cause of her morphine addiction, which is supposed to be a treatment for her "rheumatism." Now the audience sees that her morphine addiction stems from her hands, but her hands are tied up with this complicated history of the convent and of Mary's talent: perhaps Mary's rheumatism (an inflammatory disease) came from years of piano playing. We see that Mary's anxieties, her hands, and her lost dreams are all connected; this is entirely dependent on the flashback that Mary describes to Cathleen.
O'Neill adds more complexity to this story later, as Tyrone speaks with Edmund drunkenly over cards in Act 4: "The piano playing and her dream of becoming a concert pianist. That was put in her head by the nuns flattering her. She was their pet. They loved her for being so devout." Tyrone and Mary have different versions of the same story, both told in privacy. O'Neill never clarifies what the true story is, and the audience is left without a definitive answer.