Factual error: Then, as now, every recruit reporting to boot camp would be tested for illegal drugs, first by a urine test and then by a broad spectrum blood test in the case of a positive result. There is no reason for Elmo to try to hide his stash when the recruits are told they are to be tested - he is going to come up positive anyway. He may as well just say he has changed his mind and walk away. He is entitled to do that any time up to ten days after he signed on, and it happened a lot in real life!
Continuity mistake: During the graduation scene, after the platoon arrives in front of the General's stand, Bill Murray begins running backwards. The shot changes to the girls in the stands laughing, and then back to the platoon. When it gets back to the platoon, you can see that Murray is once again just starting to run backwards.
Factual error: After Sergeant Hulka escapes from the Czech border crossing, he puts out a distress call on the radio that is heard by the protagonists. In it, he says he is with the "41st Armored Division, Bravo Company," which is good enough for the heroes to identify him. But divisions are divided into brigades, which are divided into battalions, which are then divided into companies, so in the "41st Armored Division" there would be literally dozens of companies (or, more likely, "troops, " since it is an armor formation) designated "Bravo."
Visible crew/equipment: At the prison camp, during the action, watch the green barrels by the side of a building. For a brief second, you can see a camera person.
Continuity mistake: During the graduation scene, watch the band director closely when the music stops - you will see that his arms are still moving, as if he's conducting the band. (01:11:00)
Continuity mistake: When John drives the woman over the bridge in the cab, just before he stops the car there is a shot of the two occupants and behind the cab the bridge is deserted save for one red van. When the shot changes to an overhead and John stops the cab, the bridge behind them is crowded with traffic and the van is gone. (00:05:40)
Continuity mistake: When they are escaping from the prison camp, they knock over grungy, drab green 55 gal drums, which are nice and shiny inside. (01:37:30)
Factual error: The Czechoslovakian Border Guards are armed with what appears to be an Uzi submachine gun fitted with a stock. Soviet Bloc countries didn't use foreign firearms. The guards would have had Vz 58's which are visually identical to AK-47's.