Continuity mistake: In the scene where both main characters are talking in the field after the plane has crashed, Roy gives June a bottle to drink to knock her out, she drains the bottle so its empty, but a few seconds later the bottle has 1/4 left in it for her to finish drinking.
Other mistake: When June is standing on the escalator in the airport the people on the escalator with her swap from shot to shot as they change camera angles from a view looking down at her to one taken from the side. In one scene the escalator is nearly full with people, but in the next there are only two men standing behind her.
Audio problem: During their conversation on the plane, when Cameron Diaz tells Tom Cruise about the wedding gift for her sister being a car, during shot when she says "kinda like a present from my dad too you know" you can see her mouth doesn't move. It's not because of the angle of the shot either; you can see her mouthing moving from the same angle during other parts of the conversation.
Revealing mistake: In the scene in Salzburg train station, the trains that can be seen in the background are marked RENFE, which is the Spanish state-owned train operator. Also, when Fitzgerald looks at Roy through the window of the train, the station signs are "Salzburg Bahnhof", but only a few seconds later, when they are walking on the platform, the same signs are "Sevilla Santa Justa". This scene was actually shot in Seville train station.
Factual error: On the plane, June says she is restoring a '66 GTO, and flying to Kansas to get a triple deuce carburetor for it. The GTO didn't have a "triple deuce"; that was an aftermarket carb system made by Edelbrock for multiple car brands. If she's restoring the car, she'd be looking for a Tri-power carburetor set up. The Tri-power and the triple deuce are entirely different manifolds and carbs.
Continuity mistake: In the scene after June and Roy are riding on top of the car carrier truck, the CIA Director is reviewing a security tape of the diner abduction. You can hear Roy say "Pies for everybody, everybody gets pies", however, the video on the playback monitor at that exact moment shows him shooting the booth above Rodney's head. Roy didn't mention the pies until several seconds later after he kicked the server who made a noise. You can even hear the gunshot after he says "pies for everybody", then the CIA Director pauses the tape.
Continuity mistake: In the motorcycle chase, when Roy is right behind Fitz, there is a close-up shot of Roy and June in which, behind them, two beige Smarts turn up. But in the shot afterwards, the Smart has turned black all of a sudden. It can't have overtaken the first car, because the street is way too narrow.