Continuity mistake: When Don Knotts is in the old Simmons mansion, he falls down a chute of some kind. If you watch closely you will see the stunt person looks nothing like Don Knotts. It is a younger man with very straight hair.
Visible crew/equipment: When Paul Newman takes Julie Andrews up a small hill to explain that he is not going to work for East Germany you can see a shot from a distance. It is clear that this scene was done in a studio, as you can see several lights above the blue background (which is supposed to be the blue sky).
Continuity mistake: In the scene where Paul Newman seduces Shelly Winters in the nightclub, they're dancing to the sounds of a brass band, even though the musicians are all holding guitars.
Continuity mistake: On the airplane, Rock Hudson takes a pillow from the stewardess, and puts it behind his head. They cut away to the stewardess, and back to Hudson, and the pillow is gone. They cut to the stewardess again, then back to Hudson, and the pillow reappears.