Military recruiters and top brass like to repeat the refrain that the average member of the armed forces is better educated than the average American. It’s true. According to the Defense Department, nearly 94% of enlisted personnel have a high-school diploma, while only 60% of Americans do. About 83% of officers have a bachelor’s degree, in comparison with 30% of the general population.

These statistics, though, involve a bit of self-selection: Most officers have a bachelor’s degree because becoming an officer generally requires...