Revealing mistake: When the "Boat" is loaded on a "trailer", you can see that it has very little ground clearance. The wheels of the "trailer" actually are hidden behind the boards like on a parade wagon. The boat has no keel and no hull below its freeboard. Besides the question of how the boat got onto the trailer if it is practically built around it, there remains the question of how it is supposed to stay afloat without a keel or a bottom.
Revealing mistake: When Hogan, Carter and Le Beau are talking to Schultz, you can see it is an overcast day. After Hochstetter comes in and the camera cuts to them talking about Bonacelli, it goes back to Hogan. You can now see that they are on a sound stage. The reasons are that the guard isn't moving in the tower, the boys are well lit, and as Hogan and Le Beau go into the barracks, you can see differences in the painting of the sky on the back drop.
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Revealing mistake: During the main part of the plot, the guys are putting dynamite on each truck in the convoy as they fuel up at Stalag 13. It is supposed to be dark, (hence the reason the boys wear black) but there are shadows as if it were day. It's impossible for the driver in the following truck and the guard in the tower to miss seeing the boys as they go back and forth to plant charges.
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Revealing mistake: As the motorcycle and the side car split apart and drive around independently, you can see that the side car has been given two extra wheels, and some other stuff underneath as well that's not present on regular Wehrmacht BMW sidecars - probably a remote control of sorts. Also, the motorcycle Schultz rides has training wheels mounted in the shot where the side car comes off.