Chief Bromden appears to be schizophrenic because he has several of the symptoms that characterize schizophrenia, including delusions, hallucinations, strange motor behavior, disorganized speech, and lack of emotion and social interaction. He has hallucinations that fog is enveloping him. When he is being subjected to electroshock therapy, he thinks, "They start the fog machine again and it’s snowing down cold and white all over me like skim milk, so thick I might even be able to hide in it if they didn’t have a hold on me." He also has delusions that everyone who works in the hospital and in society at large is working for what he calls the "Combine," a giant machine that controls everyone's behavior. Chief constantly hears the hum of the machinery coming from the Combine. In Chapter 1, he refers to the noise of the Combine as the "hum of black machinery, humming hate and death and other hospital secrets." He is socially withdrawn and does not speak, which are also characteristics of schizophrenia. At the end of Chapter 1, he says, "But it’s the truth even if it didn’t happen." He suggests that the entire story he is about to tell might be a long hallucination.
As defined in dictionary.com, Schizophrenia is "a severe mental disorder characterized by some, but not necessarily all, of the following features: emotional blunting, intellectual deterioration, social isolation, disorganized speech and behavior, delusions, and hallucinations."
First of all, we know Chief Bromden is deaf, but he has chosen for years not to speak a word. He has maintained this charade for so long that everyone on the ward, both patients and workers, take it as a fact that he's deaf. This falls under the sympton of social isolation because he purposely excludes himself from all the other patients.
He also has elaborate and frequently occuring hallucinations that the ward is some giant mechanism that makes this at times ear piercing mechanical noise. The reader can tell he's delusional because nobody notices it but him.