Other mistake: The movie is a time piece set in the year 1963. After McMurphy jumps the fence, there is an orange/red Ford Pinto in the scene. Ford Pinto's weren't introduced until 1970. (00:51:27)
Other mistake: In the scene where the Chief smothers McMurphy, when the Chief pulls the pillow away from McMurphy's face, you can see McMurphy's tongue moving up and down even though he is supposed to be dead. (02:03:45)
Answer: It's better explained in the novel, where the Chief is the narrator. Essentially, he played deaf and dumb, so he'd be left alone by the Nurse and the staff; seeing what they did to the other patients, he figured the less he was noticed, the better. In this way, he is privy to a lot of the seedier goings-on, since they don't think he can hear what they're saying or tell anyone what he's seen.