In One Flew Over the Cuckoo's Nest, why is Bromden concerned about others finding out that he is not really deaf and mute? What happens that might...

Bromden is a character for whom silence acts as a sort of power within One Flew Over the Cuckoo's Nest. While he has lived on the ward silently for a great period of time and cannot make deep connections with its residents, he nevertheless experiences a form of personal power in the ability to remain relatively unnoticed by staff. As such, he overhears their conversations and meetings and is privy to information he might otherwise not have. Yet, when McMurphy proposes a fishing trip within the novel, Bromden feels a sense of deep inner conflict. He wants to put his name on the sign-up sheet for the boating trip, but realizes that doing so will signal to both patients and staff that his deafness/muteness is not genuine. Kesey writes of Bromden's inner-debate:

"I didn't have the money, but I kept getting this notion that I wanted to sign the list. And the more [McMurphy] talked about fishing for Chinook salmon the more I wanted to go. I knew it was a fool thing to want; if I signed up, it'd be the same as coming right out and telling everybody I wasn't deaf" (Kesey 118). 

A major conflict, for Bromden, lies in his desire to form connections to other people and his desire to retain some sense of agency on the ward. While signing up for the trip means the loss of one “freedom” (his ability to go unnoticed), it gives him another (meaningful experiences outside the ward). The complication of this definition of freedom, as you might conclude, is that it proposes a sort of power lies in being unnoticed and/or unheard, which, for a character whose childhood experiences of being unheard were profoundly damaging, is troublesome.

Kesey, Ken. One Flew Over the Cuckoo’s Nest. New York: Signet Books, 1962. PDF file.