From Green Berets to the S-1 clerk and everyone in between, every soldier has something in common with each other: they’ve all started their military careers at the U.S. Army’s Basic Combat Training.

VIRGINIA BEACH, Va. — More than 40 feet up the tower at Fort Jackson, Army recruit Robert Smith missed setting his feet on the 4-by-4 plank from which he was to start rappelling down. “I said: ...