Continuity mistake: Throughout the movie the outside shots of the B 52 show a shadow on the ground. That shadow isn't of a B 52 jet bomber but that of a B 17 WWII prop. It isn't a B 29 either, for the cord of the B 29 wing is thinner than a B 17s; Also, the wings and fuselage are much longer. The most obvious clue is the shape of the B-17 vertical tail- it is unique to the B-17, and much different from that of the B-29.
Continuity mistake: When Mandrake and Ripper are in the latter's office and he is firing out of the window, Mandrake holds a chair in front of him for protection. In the next shot from behind them, the chair is nowhere near him. (00:52:10)
Continuity mistake: Just before Ripper kills himself, the cigar's shadow across his face indicates bright lights at the back of his office, away from the windows. When Ripper and Mandrake are seen standing together, bright lights are coming from the other direction. The direction of the shadows alternates between the close-ups and the wide shots.
Continuity mistake: During Dr. Strangelove's last delivery in the war room the Russian ambassador is standing next to his wheelchair, looking straight and grim. Strangelove develops some rather macho ideas about the ratio of women and men in the mine shafts after, and the ambassador turns to him and says with a smile, "You have an astonishingly good idea." but when the camera angle changes he is looking as straight and grim as before. (01:27:55)