Deliberate "mistake": There is a huge security breach in the scene with Robin Williams and Christopher Walkin walking at night. The Security SUV limousine is more than 30 metres away from them. It should be pacing them in case of an emergency and be right next to them on the sidewalk. It appears the reason it is so far back is because at the camera angle the scene is shot at, it makes for a good background element which you would not see if the Secret Service followed protocol.
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Mistakes
In the scene where Laura Linney is talking to Christopher Walken at the Thanksgiving party, he is sitting in his chair smoking a cigarette next to a patio door. During their conversation, while the camera is facing Christopher Walken, he is holding his cigarette in his right hand. When the camera changes to face Laura Linney, his cigarette is visible over his left shoulder and is held by his left hand. When the shot changes back to face him, it's back in his right hand. The shots seem to move too quickly for him to have switched hands between them. See more...
Man of the Year mistakes
Continuity: Right at the start of the paintball game, we see that one of the players does not have a hopper (paintball holder) on the top of his marker (paintball gun). And he is firing blanks.
Continuity: Right after the debate, we see Tom Dobbs and Jack Menken walking along the sidewalk talking. From the wide shot we see that they walk past the mail box on the sidewalk. Then when the camera cuts in to the close up, we see then walk past the same mail box again.