Other mistake: Near the end, Harry Lime is trying to escape through the sewers of Vienna. There is an opening to the street; Harry puts all his fingers through the holes, trying to lift the metal plate, but Harry has a pistol in his right hand; there is no way he can poke all his fingers through with his right hand with the pistol in it.
Trivia: Orson Welles' line "...you know what the fellow said... In Italy for thirty years under the Borgias they had warfare, terror, murder, bloodshed - but they produced Michelangelo, Leonardo da Vinci, and the Renaissance. In Switzerland they had brotherly love, five hundred years of democracy and peace, and what did that produce? The cuckoo clock." was not in the script. Welles just wanted to say it, so he did, much to the anger of the original author. He thought it was a terrible piece of writing but it has turned out to be the most memorable line in the entire film.