Corrected entry: In the scene in the club when Jude Law is playing the saxophone, you never actually see his fingers move while he's playing.
Corrected entry: The scooter Jude Law rides at the beginning (a Vespa), when he's first seen kissing another girl, is a modern version, which could never have been around in the 50s.
Correction: Dickie's Vespa is a 150GS first introduced in 1955 so it is completely possible to be in the time frame.
Corrected entry: When the American inspector shows Tom the license plates found in Dickie's luggage (near the end of the movie), the larger plate has '2007' on the bottom. One of the numbers must be facing the wrong way, because no matter which way the plate is turned, either the 2 or the 7 is always upside down.
Correction: It is actually '4002', so it is the correct way round, and the same on both plates (except the 'EE' comes at the end on the small plate).
Correction: Law actually learned to play the saxophone for the film, and if you watch closely during the club scene in Naples when both Damon and Law perform, his fingers do indeed move in time with the proper notes.