When John wins the chess game and Paul is called over, Paul gives John's opponent advice on how he could have won. Paul couldn't have known this so quickly without observing the game or the board for at least a few minutes.
The Ambassador Hotel's lobby scene, in which John Casey and Nelson talk, shows Casey wearing a quartz watch. It was June 1968. The world's first commercial quartz wristwatch, the Seiko Astron, would only be introduced in December 1969.
The slogan "The once and future king" on the kitchen wall was written by Edward (Laurence Fishburne), who was left handed. As such the first couple of words are rubbed slightly by his sleeve. In the final scene we see a blood-spattered slogan, this time written perfectly without the faded first couple of words.
When Jimmy and Cooper are sitting in the coffee shop, Cooper holds up the camera and the lever on the bottom is pulled back. Then, when the shot comes back to them, the lever is forward and Jimmy reaches over and pulls it back.
The Christian Slater character violently grabs William Macy between the first and second buttons on the tuxedo shirt with a completely bloodied hand. Later at the ambulance, Macy's perfectly buttoned shirt has a only few minor specks of blood, should be a bloody mess.
The credits include the closing speech detailing the speech as "Robert F. Kennedy's speech, 'On The Mindless Menace of Violence.' The credits say it was delivered in Indianapolis, Indiana on April 5, 1968. This is incorrect; Robert Kennedy gave a speech on Martin Luther King, Jr.'s death in Indianapolis on the previous day but gave the speech presented on the recording at the City Club of Cleveland in Cleveland, Ohio.