Audio problem: Hogan knocks out his pipe against the Prussian helmet on Klink's desk. It makes a tinny, metallic sound. The Prussian Pickelhaube was made from compressed leather, not metal.
Continuity mistake: When Newkirk calls "Heil Hitler", everyone in the lobby salutes. A second later, in the closeup of Hogan and Newkirk, everyone's arms are suddenly back by their sides, much too quickly to be real.
Continuity mistake: When explaining the "experiment" Lebeau has going, the multi-chambered tube is on the left (from the camera's POV) and the long tube with the coil is on the right. As the men up to leave after Carter comes in, the tubes are reversed.
Answer: The core POWs regularly escaped the prison camp but they made it their mission to conduct espionage and commit sabotage in the surrounding German territory. They also collaborated with different underground resistance groups and used a network of secret tunnels to help prisoners from other POW camps to escape, who then relayed vital information back to the Allied forces. Hogan and his men maintained the illusion that Stalag 13 had never had any prisoners escape in order to avoid their covert operations being shut down. Being that the prison camp is set in Germany during WWII, there were no American embassies.
raywest
Hogan has mentioned to different characters that they are actually stationed at Stalag 13 to help allied soldiers and prisoners from other Stalags to escape Germany.