Plot hole: At the end of the film the boss says that he and Miss Fitzgerald are going to go on holiday together to Paris (talking about the actual city, not the secret code they have invented). However, the film is set in 1940 and Paris would have been an unusual place to take a holiday at this time, considering France was in the middle of a war... Pretty dangerous for civilians even though America was still neutral at the time.
Visible crew/equipment: When Woody Allen arrives at the mansion where the second robbery took place and starts talking to his friend who's investigating the crime scene, one can see the mic's reflection on the big glass door of the house.