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.
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.
Other mistake: During the graduation scene, if you watch as the platoons are marching by, you can see that several members are not in cadence as they should be. This also includes the men walking between the platoons that are supposed to be setting the cadence and they are out of step with the platoon in front and behind them.