Stupidity: Major Franklin is struggling up the side of the cliff while the rest of the group are on top of the cliff checking their equipment, then he slips and slides down the cliff side, breaking his leg. After seeing this, the group members reach for a long length of rope that was used to bring up the other members and the supplies. The commandos could have used the rope to help Major Franklin climb up and prevent him from falling and injuring himself.
Stupidity: When they are preparing for a German assault towards the end of the movie, Miller explains to them the concept of the sticky bomb and adds, "If you have a better idea of knocking the treads off a tank I'd like to hear it." Um... The bazooka? They had eight rounds for it, and it could have easily been used to disable the treads. And the men wouldn't blow themselves up with it. While it couldn't destroy a tank by penetrating the armour, it's certainly strong enough to disable the tracks.
Stupidity: As the position of the Tiger was perpendicular to the Sherman platoon's line of march, and they were advancing across totally open country with firm ground all around, it didn't matter a hill of beans whether they shot the first or the last tank in the column.
Stupidity: The lack of security at the château is not plausible. The Germans typically would have layers of protection around headquarters or other sensitive areas. In the movie, there are two middle aged guys guarding the road into the Château. After that there are no checkpoints, no patrols, and no controlled access points into the castle. It makes no sense that dozens of officers would be left vulnerable to an attack by French resistance fighters, let alone an Allied airborne assault.
Stupidity: Staff Sergeant Duquesne is tossing and flipping rifle grenades before affixing them to his rifle to launch against the German tanks. A battle-hardened no-nonsense sergeant like Duquesne would never waste valuable time showing off, and would definitely not flip or toss around explosive ordnance like rifle grenades.