Deliberate mistake: In all the shoot-outs, the bullets that hit the cars spark, which bullets do not do, and leave no hole, dent, or even a small scratch. They do, however, break the glass.

The Departed (2006)
1 deliberate mistake
Directed by: Martin Scorsese
Starring: Alec Baldwin, Jack Nicholson, Leonardo DiCaprio, Mark Wahlberg, Martin Sheen, Matt Damon
Other mistake: When Costigan drags a handcuffed Sullivan into the elevator, Costigan slams Sullivan against the wall and his left arm is visible and not in handcuffs. (02:13:05)
Surveillance Guy: Who the fuck are you?
Dignam: I'm the guy who does his job. You must be the other guy.
Trivia: This was Martin Scorsese's first film to win Best Picture. And his first Oscar for best Director. Martin also said that this was the first movie he ever made with a plot.
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Answer: The FBI and police are separate organizations that do not share full information. Only the FBI and Costello himself knew about the informing, and the FBI can only do so much to stop Costello being caught without drawing attention to their relationship. The police and Costello's gang are fighting a separate battle, hence the two moles.