Because it wasn't clear (and still isn't, if you study it closely) that deterrence actually "works" as a long-term stable strategy. There were numerous instances where peace and stability were threatened in and around the 1960s: the Cuban Missile Crisis stands out as the worst of them, but consider also the Berlin Crisis, the Suez Crisis, the Taiwan Strait Crisis, the Vietnam War. Any one of these had a chance of "escalation," where one small, conventional act could become a ladder, through deliberate choice or accident, towards the use of nuclear weapons.

The Cuban Missile Crisis is a great example of all of these possibilities. Two episodes:

In the middle of the crisis, a security guard at a Duluth air force base on night duty thought he saw a spetznaz agent sneaking into base, presumably to sabotage it before a Soviet attack. The guard shot at the supposed saboteur and sounded the alarm, which also rang alarms at nearby bases with nuclear missiles. At one such air force base, the alarm was wired incorrectly, and instead of signally a sabotage, who started to scramble their nuclear-armed interceptor planes. The saboteur turned out to be a bear.

In another incident, a Soviet submarine near Cuba was being "signaled" to rise to the surface with depth charges by an American destroyer. They thought (not inappropriately) that they were under attack and that war had broken out. They had nuclear-armed torpedoes and, per Soviet protocol, the three officers on board took a vote as to whether to use them. One officer rejected the idea, the other two did not. That one vote kept them from doing it.

In the first case you have misunderstanding (bear = agent) and malfunction (wrong alarm wiring), and that led to nuclear-armed planes scrambling and looking for enemies. In the second you have misunderstanding (signal = attack) almost leading to semi-rational nuclear use. There are many other such "close calls," often combinations of misunderstanding, malfunction, fear, etc. You only need one of those things to align correctly for things to potentially escalate — both the command and control systems of the USA and the USSR were based around the idea that once a nuclear war started, you had a "use it or lose it" situation where you had to use your nukes before they got destroyed by the other guy's nukes.

Misunderstanding can be profound. During the Crisis, President Kennedy's military advisors were pushing him to do a land invasion of Cuba, believing that no nuclear weapons were on the island yet. Only in the post-Cold War did they learn that they were wrong in that belief: the Soviets had already moved hundreds of warheads to the islands, many of them tactical nuclear weapons, and would have used them to repel an American invasion, leading to incredible casualties and further escalation.

Add to it that deterrence relies on rationality, and not all leaders of nuclear nations are "rational" by any means. China in the 1960s was considered as fractious and bizarre as North Korea is today, and set off their first bomb in 1964. They were ideological and belligerent and isolated, and believed that with their huge population they could weather a nuclear attack and repopulate the world — the scariest sort of people to have nukes, because they might believe (rightly or wrongly) that nuclear war was a good path for them.

As General Lee Butler put it, as head of Strategic Air Command and while reviewing US war plans at the end of the Cold War: "I came to fully appreciate the truth … we escaped the Cold War without a nuclear holocaust by some combination of skill, luck, and divine intervention, and I suspect the latter in greatest proportion."

If you're interested in close study of these things, check out Scott Sagan's The Limits of Safety, or for a more popular take, Eric Schlosser's Command and Control.