Most historians agree that the world came closest to a nuclear war with the Cuban Missile Crisis of October 1962, when President John Kennedy discovered that the Soviet Union had installed nuclear missiles in Cuba and warned Soviet leader Nikita Khrushchev to remove them—or else. For two weeks messages flashed back and forth between Moscow and Washington. As tension mounted and U.S. forces, including nuclear-loaded B-52s, were placed on high alert, the Soviets blinked, agreeing to dismantle the sites and ship their missiles back to Russia.

But now a British popular historian and a veteran American reporter, in separate books, contend that the most dangerous moment of the Cold War occurred in early November 1983, when the Soviets nearly launched a nuclear attack against the West because they thought that NATO was planning a first strike under the cover of a war-game exercise. Drawing on documents obtained through the Freedom of Information Act and first-time interviews with national-security experts, Taylor Downing in “1983: Reagan, Andropov, and a World on the Brink” and Marc Ambinder in “The Brink: President Reagan and the Nuclear War Scare of 1983” conclude that, without most of us knowing it, the world came close to war 20 years after the crisis in Cuba.

Among those who were not aware, at the time, of a possible nuclear Armageddon were President Ronald Reagan, on an official visit to Japan and South Korea, and Mikhail Gorbachev, then a Politburo member, who now recalls that he celebrated the October Revolution as planned and knew nothing of a war alert. On Nov. 18, 1983—a week after the war-game exercise ended—Reagan was briefed on the Soviets’ startling belief that a first strike from the U.S. might soon be launched. He wrote in his diary: “I feel the Soviets are so defense-minded, so paranoid about being attacked, that without being in any way soft on them, we ought to tell them no one here has any intention of doing anything like that.”

Mr. Downing and Mr. Ambinder stress that the paranoia of Soviet leaders, especially that of Yuri Andropov, the general secretary of the Communist Party and former KGB head, was reinforced by a global intelligence operation that the Soviets had undertaken around the same time. It was named Project RYaN, an acronym taken from the Russian words meaning “nuclear missile attack.” KGB agents in the U.S., Britain, other NATO countries and Japan had been tasked to look for indicators that a nuclear attack was being planned. Be on the lookout for more aggressive language by political leaders, they were instructed, as well as any heightened military alert.

Trained to give their bosses what they wanted, KGB agents responded so enthusiastically that, before long, they had identified 292 different “signs of tensions,” strengthening the suspicions of Andropov and other hard-core Politburo members. U.S. intelligence officials, for their part, played down the elevated Soviet alert when they learned of it, because they couldn’t believe that the Soviets would seriously think a U.S. strike was imminent.

According to Mr. Downing and Mr. Ambinder, the Soviet worry over U.S. militancy had been intensified by recent events: Reagan’s description of the Soviet Union as an “evil empire,” made in a speech in March 1983; NATO’s decision to deploy Pershing II and Cruise missiles in Western Europe, a response to the Soviet placement of SS-20s aimed at targets in NATO countries; and the Reagan administration’s commitment to the development of the Strategic Defense Initiative, the high-tech missile shield. But the immediate cause of the Soviets’ fear was Able Archer, as it was called: an exercise conducted annually to test, among other things, NATO’s response to a Soviet tank-led invasion.

While NATO engaged in make-believe warfare, Moscow tracked suspicious moves, like increased communications between Washington and London, and a U.S. field commander suddenly transferring his headquarters to another mobile convoy. Able Archer’s test-scenario presumed the possible NATO use of tactical nuclear weapons; the Soviets noticed the participation of B-52s for the first time in the exercise. The B-52 presence meant one thing to the Soviets, writes Mr. Ambinder: nuclear strikes.

At times, Mr. Ambinder’s book reads like a Tom Clancy novel, as when Soviet Capt. Viktor Tkachenko, deep in an ICBM bunker, and 26-year-old U.S. Capt. Lee Trolan, in charge of a dozen nuclear weapons, are described handling the escalating tension. We watch Capt. Trolan guarding nuclear weapons with six men for every ten he should have had, working seven days a week. “If you asked me then whether I thought we were going to have a shooting war with the Warsaw Pact . . . I would have said, yes.” As for Capt. Tkachenko, he had heard rumors that “the Americans would wait until the eve of a major Soviet holiday,” when ordinary Soviets were relaxed and happy, “to launch World War III.” He returned again and again to the critical question: Is “Able Archer 83” a normal military exercise or the incomprehensible—preparations for a nuclear strike?

With Soviet forces on high alert, Mr. Downing takes us to the Kuntsevo Clinic outside Moscow, where a critically ill Andropov is receiving dialysis treatment. It is the evening of Nov. 9. Three men could give the launch order—Andropov himself, Defense Minister Dmitri Ustinov or Chief of the General Staff Nikolai Ogarkov. How will the day end: in a nuclear attack on the Soviet Union, or will Moscow strike first?

Mr. Downing speculates that, in the end, Andropov and the other Soviet leaders, conscious of the terrible loss of Soviet life during World War II, “did not want to push the nuclear button unless they absolutely had to.” Separately, they were uncertain that RYaN was correct about NATO’s war preparations. And there was the reassuring report, as Mr. Ambinder recounts, from a reliable Soviet spy at NATO headquarters that “nothing was happening.”

By the time dawn broke on Nov. 10, Moscow had not received any alarm of enemy launches, and Soviet forces were ordered to stand down. Mr. Downing writes that “probably the most dangerous moment of the Cold War passed.” But was it truly that dangerous? We can say with certainty, thanks to Mr. Downing and Mr. Ambinder, that both Soviet and American intelligence got it wrong—the Soviets displaying a scary degree of paranoia and the Americans unable to accept the truth of their paranoia. But there is still a great deal that we don’t know—not least about what Andropov was thinking and how close he came to launching a nuclear attack. Until the Kremlin opens its files, we are in a swamp of speculation. In any case, the authors are to be congratulated for a splendid job of research about a critical event in the Cold War that other historians have overlooked or underplayed.

It should be added that Ronald Reagan had been talking about doing away with all nuclear weapons since the 1960s. He often remarked that the U.S. policy of Mutual Assured Destruction (MAD) reminded him of a Mexican stand-off: There had to be a better way. As Mr. Ambinder writes in an epilogue: “Reagan realized that only presidents can truly understand the entire dimension of the nuclear problem, only presidents have the power to elevate global stewardship and sovereignty above patriotic partiality, and therefore only presidents can keep the world from the ash heap of Armageddon.”