Other mistake: In the scene where Dillinger and his gang raid a bank and take the bank manager and a teller hostage, they come out shooting with machine guns and jump into and on the running boards of the getaway cars, all through this encounter the bank manager shows no concern for his own safety, no reaction whatsoever to flying bullets and loud gunfire, very strange.
Continuity mistake: When John gives Billie her mink coat, the amount of wrap paper coming out of the corners of the box differs between the front and back angles.