This is a movie that will leave you thinking, is he or isn't he? While many people have complained about the ambiguous ending, it gives room for the audience to think and interpret it from the signs. This is my interpretation and theory, and I believe it is very sound. 

First, here is the plot. One day, Prot (Kevin Spacey) suddenly appears in the midst of a busy train station. After attempting to help a woman from muggers, he is arrested and sent to Bellevue, and later transferred to Dr. Mark Powell's (Jeff Bridges) hospital. Prot freely talks about how he came from the planet K-Pax and is here to do a report on Earth. Naturally, he is classified as a looney and is locked up in a low security level ward. He befriends the other patients and quickly convinces them of his story. In fact, he tells them he can take one person back with him. Soon Dr. Powell is beginning to question Prot's insanity and as the plot progresses, it is harder and harder to prove Prot wrong. 

In the end, Dr. Powell learns that Prot is Robert Porter, a smart man who worked in a slaughter house in New Mexico, when his wife and daughter were murdered. Prot claims that he plans to leave for K-Pax at a precise time on the fifth anniversary of the murders. The time comes and goes, and Powell finds Prot or Robert in a catatonic state. One of the patients is missing and is never found again. 

On to my theory: Prot is not crazy. Prot is from the planet K-Pax. Robert Porter is a friend of Prot's. On one of Prot's previous visits to Earth, Robert was a child, learning about the constellations from his father. Prot and Robert became best friends. After the death of his family, and Robert slaying the murderer, Robert decides to commit suicide. Prot is on K-Pax at the time, but he rushes back (in multiples of light speed) to stop Robert. So Prot takes over the thinking of Robert, taking over his body so to speak. Awhile later, Prot (still inhabiting Robert's body) comes to New York and is locked up. When he leaves for K-Pax on July 27, he takes Bess with him, but he must leave Robert's body behind. Hence, we now have the catatonic Robert.

Explanations or clues:

PROT'S KNOWLEDGE OF THE PLANETS AROUND LIBRA: As Prot says, any child knows about their own system. Only a few people on Earth had an inkling about the system, and Prot added to their knowledge. Robert Porter, while smart and knows his constellations, was no astro-physicist. 

DR POWELL FEELS CHOSEN BY PROT: It's true. Prot could've left the institute at any time, but chose not too. Prot's last words to Dr. Powell are, "Now that you've found Robert. Take care of him." Prot was probably staying there to ensure a place for Robert when he left. 

SUPER-HUMAN ABILITIES: It was a proven fact that Prot had a significantly higher sensitivity to sunlight and could even see UV rays. Prot provides his usual K-Pax reason, but there was no Earthly reason given. It was also mentioned that Prot was given extremely high doses of medicine with no effect.

HELPING THE PATIENTS: Throughout the movie, you see Prot helping the other patients (or the patients discussing Prot's suggestions) to a better sane world. While the psychologist balk at Prot's help, Dr. Powell talks at the end of the movie about the new lives the patients are leading.

THE SPRINKLER SCENE: The sprinkler is associated with Porter's home in New Mexico (which is ironically where Roswell is). Does Porter associate the sprinkler with the death of his family? Is her trying to protect Powell's daughter from that horrible fate? I have a different point of view. It is assumed that Porter had committed suicide in a river. Indeed, Prot (under hypnosis) is greatly concerned about Robert. I believe that Prot associates the danger with the water, not the sprinkler. He is trying to protect the girl from watery death, like he did Robert.

HYPNOSIS: Dr. Powell feels that under hypnosis he can uncover Prot's true identity, but while Prot gives plenty of information, he never gives up his Prot persona and claimed homeland of K-Pax.

THE SECURITY CAMERA: Why would the camera just give out for no apparent reason? 

MYSTERIOUS DISAPPEARANCES: First, Prot disappeared for a few days to his trip up north. Funny, the patients are not at all alarmed. Indeed, they try to reassure the doctor. Sometimes I think they understand better than the staff. Naturally, the most telling disappearance is of Bess, the winner of the essay contest. Not only does she disappear at the approximate same time, but the only remaining clue is her winning essay.

This is only my humble theory. You can conclude what you want. You can take Dr. Powell's theory, that Prot was merely a persona taken up by Robert Porter to mentally defend himself from reality, but that would be giving up too many plot holes.

Sentiment: positive