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.
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: In several episodes, the fence around the cooler is visible, and hanging on it, a sign saying "Eingang verboten" meaning "no entry." The correct German term would be "Zutritt verboten." In German, "Eingang" is the opening where you enter a building, not the act of entering one.
Factual error: When the SS guard is calling Sergeant Shultz after seeing the burning fuse, he addresses Shultz as "Sergeant", rather than the proper German word for Shultz's rank, "Feldwebel."
Continuity mistake: Hogan dumps a bucket of water on Barnes and Davis each, to make it look like they were the ones who hid in the tanker. They get big wet patches on their uniforms from it, but aren't completely wet through and through - which is to be expected, after all it's just a small bucket of water each. After the next cut however, they are completely soaked like they went swimming.
Plot hole: Hogan has his men play romantic music over the radio. Earlier they were afraid of the radio in the observation post being detected. Elsewhere in the series, they try to avoid sending too long to avoid being homed in on - which is correct. Now, if they were to play a whole record at once, wouldn't the homing devices pick up on the transmitter at the camp?
Factual error: While the barracks were reasonable replicas of the real POW barracks, there was one main difference. The original buildings were elevated to ensure any escape attempts could be stopped. In this series, the barracks are flush on the ground, which makes the tunnel access possible and easy.
Factual error: Baker and Kinchloe, the radio experts of the troupe, often use Morse code to communicate by radio. When they do, they hammer the Morse key in different intervals, but always just barely tap it. Morse code is made up of "short" and "long" beeps. To produce a "long" in Morse code, you have to hold the key down three times as long as you would for a "short". A tap would be a "short" - the beeps they are sending are spaced long and short, but that's not how Morse code works.
Character mistake: In several scenes all over the show (though not always!), characters use the command "Raus!" to send somebody away from somewhere. Raus, short for "heraus" literally means "out" in the sense of "out of a building", which would be "aus einem Gebäude heraus" in German. Used as a command, it always means "get out", never "get away from there" or "get lost", in other words, the addressee must be inside of somewhere to be ordered "Raus." One example would be Schultz sending the prisoners away from general Burkhalter's car in S05E13.
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.