Continuity mistake: Solo and Illya park their big Chrysler convertible in front of Del Floria's and go inside. The shot then cuts to the tailor shop's interior with a view of the street through the window, and the car they just parked out front is nowhere to be seen. (00:00:20)
Other mistake: Vladik, incorrectly, tells Donfield on the phone to stand at the hot dog stand at 3 pm Monday. It should have been 11 am. Donfield in the "meet" scene refers to his 11 o'clock feeding" and Mr. Smith tells Solo (Donfield) his next meeting is between one and two pm. This would be impossible if the original meet was at 3 pm.
Continuity mistake: When Napoleon and Illya are trying to escape from the steam room, and Illya accidentally drops the key to the door, Napoleon points to where he thinks the key is with his right hand, with his left hand resting on the bench above him. When they finally find the key, Napoleon's left hand is pointing at its location. (00:40:25)