Factual error: The German agent pretending to be an American pilot is wearing a one piece flying suit from the 1960's when he is questioned by Hogan and the British escapee in the barracks. It even has the US Air Force logo on the sleeve, which didn't come into use until 1947, after the war.
Revealing mistake: When Schultz checks the barbed wire to see if it is fixed now, he noticeably wiggles the vertical wire, but only barely touches the horizontal one - because it is rigged to come apart, and will again if he handles it too roughly.
Other mistake: When the boys are listening to Klink's office, the lid is on the speaker instead of being off.
Continuity mistake: When Newkirk brings his hat loaded with candy, he hands it to Hogan with the badge on the top on Hogan's left. When the camera angle changes, the hat has been reversed.
Revealing mistake: When the boys are working on the fence, when the camera is on Hogan, you can see a modern metal building in the distance.
Answer: It's a comedy, not a documentary.
stiiggy
Perhaps it was counterfeit. There are numerous episodes where they deal in counterfeit monies.