Other mistake: When Raymond shoots Senator Jordan, his first shot penetrates the carton of milk that Jordan is holding in his left hand. From Raymond's perspective, that carton of milk is positioned over Jordan's upper left arm; so, the first shot should have merely wounded Jordan in the left bicep, hardly a mortal wound. But Jordan's eyes roll up and he collapses as if shot in the heart.
Character mistake: During the press conference the Secretary of Defense yells at Senator Iselin "Where is the Sergeant-at-Arms? Throw that man of out here!" The problem is that "Sergeant-at-Arms" is a congressional post and that is not a position that would exist at the Defense Department (the Pentagon), nor would the Secretary of Defense hold a press conference in the Capitol (where someone who has that title would be).
Answer: The shot in question occurs when Sinatra's character, Marco, holds up a deck full of queens while trying to deprogram Raymond. On the DVD commentary track, the director, Frankenheimer, acknowleges that the scene was out of focus, and that though Sinatra supplied several other takes of the scene, the other takes weren't nearly as good, so he went with the flawed one. Later, Frankenheimer was praised by critics for the unfocused shot showing Raymond's disturbed perceptions.