Chief Bromden has not really become deaf and dumb; he simply feigns his deafness in order to gain knowledge about the Big Nurse and the orderlies; for, they talk in front of him, assuming that he cannot hear:

They don't bother not talking out loud about their hate secrets when I'm nearby because they think I'm deaf and dumb.  Everybody thinks so, I'm cagey enough to fool them that much.

Fully aware that his being deaf is an act, Bromden narrates further in Chapter 2, as he recalls first meeting McMurphy,

But then's when I remember thinking that he was laughing because he wasn't fooled for one minute by my deaf-and-dumb act....

Later, in the first chapter of Part III, Bromden lies in bed the night before the fishing trip and ponders his charade of being deaf, of not "letting on I heard what was being said," and he wonders if he can now act any other way since his pretence has occurred because people have thought he has been too insensate to hear or see or speak about anything.  Recalling the words of his father,

Papa says that if you don't watch it, people will force you one way or another, into doing what they think you should do, or...into doing the opposite out of spite,

Bromden's understanding underscores Kesey's theme that society demands conformity and, as Ralph Waldo Emerson expressed it in Self-Reliance, is "in conspiracy against the manhood of every one of its members." Ironically, then, Bromden feigns being deaf in order to defeat what he feels is the conspiracy against his manhood and individuality.  So, he speaks in order to warn McMurphy,

"They can't have somebody as big as Papa running around unless he's one of them.  You can see that."

He warns McMurphy of the Combine that will use various means to subjugate him.

The answer to this question can be found in Part III, when Bromden experiences his first memory of his past, and in particular the way that he is treated by the white men that come and try and buy the land that is his father's inheritance as chief of his tribe. What Bromden remembers in particular is the way that they ignore him and do not even look at him, denying his humanity. Note what Bromden sees when he looks at the white men he has just talked to:

And, almost, see the apparatus inside them take the words I just said and try to fit the words in here and there, this place and that, and when they find the words don't have any place ready-made where they'll fit, the machinery disposes of the words like they weren't even spoken.

Chief Bromden believes he is deaf and dumb because he is acting upon how he is treated by society, as represented through this memory of how the white men treated him. Because his words "don't have any place ready-made where they'll fit," it becomes clear that Bromden internalises this belief and sees the logical extension to be that, as others seem to not be able to hear what he says, he cannot talk at all. Bromden therefore adopts his deafness and dumbness as a response to the way that society at large treats him.