Every enlisted man dreams of it: pulling rank on the military’s highest brass. The heroics of John F. Kennedy, lieutenant, junior grade, in the South Pacific after his PT‑109 was sunk in 1943 eased his way, 17 years later, to being elected the nation’s commander in chief. In the White House, he fought—and defeated—his most determined military foes, just across the Potomac: the members of the Pentagon’s Joint Chiefs of Staff. “Here was a president who had no military experience at all, sort of a patrol-boat skipper in World War II,” Joint Chiefs Chairman Lyman Lemnitzer later said of Kennedy. Mutual respect, from the first, was in short supply.

In comparison, Nikita Khrushchev was a pushover, at least during the events that brought President Kennedy’s most-notable achievements. By persuading the Soviet leader to remove missiles from Fidel Castro’s Cuba and agree to a ban on nuclear tests in the atmosphere, underwater, and in outer space, Kennedy avoided a nuclear war and kept radioactive fallout from the air and the oceans, thereby earning the country’s enduring regard for his effectiveness as a crisis manager and negotiator. But less recognized is how much both of these agreements rested on Kennedy’s ability to rein in and sidestep his own military chiefs.

From the start of his presidency, Kennedy feared that the Pentagon brass would overreact to Soviet provocations and drive the country into a disastrous nuclear conflict. The Soviets might have been pleased—or understandably frightened—to know that Kennedy distrusted America’s military establishment almost as much as they did....