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.
The Producers (2005)
1 deliberate mistake - chronological order
Directed by: Susan Stroman
Starring: Will Ferrell, Matthew Broderick, Uma Thurman, Nathan Lane
Continuity mistake: When Max Bialystock is singing "Along came Bialy", and the old ladies start coming out of the apartments, the first lady who comes out isn't there in the next shot.
Mr. Marks: Do I smell the revolting stench of self-esteem?
Trivia: Mel Brooks "composed" the music for the Broadway show (which would go on to become this movie) by humming tunes into a tape recorder. Someone who was musically literate would then write it down on staff paper.