Deliberate mistake: When the police come into Max's office near the end, they force Franz Liebkind to drop his gun. Immediately after doing so, Franz cringes in pain, implying that the gun was dropped on his foot. However, the gun was dropped about 2 feet from where his foot would have been. It would have been literally impossible for the gun to have hit his foot.
Deliberate mistake: In the ambulance scenes, John Goodman is given oxygen via a nasal tube, probably so he could tell his story. In real life, a mask covering the nose and mouth would be used for maximum effectiveness in such a critical situation.