The Battle of Stalag 13 - S2-E5
Revealing mistake: When the scene transitions from Klink and Hogan to a night shot outside of the barracks, you can see palm trees above the roof line. Not normal for the area that's supposed to be represented.
Revealing mistake: As Hogan goes through the hole to the ladder, the "rock" he brushes against flexes for a second. Likewise, when Kinch climbs through, the "rock" under his hand flexes and mashes down as his weight is on it.
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.
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.
Hogan and the Lady Doctor - S2-E20
Revealing mistake: As the truck pulls into the laboratory yard, the buildings of the city surrounding the studio backlot are seen.
Revealing mistake: As Burkhalter's car comes into the camp, it's easy to see that it's a re-use of stock footage.
Heil Klink - S2-E22
Revealing mistake: After Browner gets out of the staff car and is going inside, as the camera pans to the left following him, you see the Desilu Studios water tower.
Answer: It's a solitary cell. Steve McQueen, star of 'The Great Escape' is known as the 'Cooler King'.