Plot hole: In several occasions throughout the story, e.g. S4E18, Hogan attends parties at Stalag 13 wearing an immaculate US dress uniform. Where did he get that? Dress uniforms are not part of the usual kit a bomber crew brings along with them on their missions.
Factual error: The "Blue Baron" tells the dancer that the Kaiser gave him a certain medal. In fact, the medal he points to (and the girl fondles) is a WW2 repeat badge to the Iron Cross first class, instituted in 1939 to denote presentations of the Iron Cross first class to personnel who had already received it in WW1. He may have received the original Iron Cross from the Kaiser, but by the time he had a chance to receive that repeat badge, the Kaiser was long through handing out medals. (00:15:10)
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.
Continuity mistake: When Helga brings in Klink's dinner and sits it down, Klink takes off his monocle. In the close up, it's on and his hands are folded, then he points to a chess piece, contemplating a move. In the wide shot, he's holding his monocle, and puts it on.
Factual error: In the tunnel Hogan loads a magazine into a US issue .45 pistol and points it at Myra, but he never chambered a round and didn't cock the hammer on the pistol. The .45 pistol in WW2 was single action. For a single action pistol to fire, a round needs to be chambered and the hammer has to be cocked manually before the trigger is pulled and the pistol to fire. (00:22:00)