During the Cold War the risk fluctuated all the time. The closest we ever came to nuclear war would have been in 1953, 1962 and 1983.

1953 saw the death of Stalin and khrushchev came into power. Eisenhower becomes President of the United States, and both leaders' start testing weapons within months of each other. The Rosenbergs trial was coming to a close, and the USA was utilising 'Duck and Cover ' videos in schools to prepare for a nuclear strike.The doomsday clock stood at 2 minutes to Midnight at this time.

1962 Was the year of the Cuban Missile Crisis. The US had installed missiles in Turkey, too close for the USSR. The USA had far more weaponary and the khrushchev was concerned. Fidel Castro, in light of the failed Bay of Pigs invasion, appealed to the USSR for protection. Khrushchev and Castro agreed that Cuba would house Soviet Missiles as a deterent to the US, being a mere 90 miles from US mainland. With the public unaware at the time, US President Kennedy and Khrushchev entered into brinkmanship trying to resolve the matter. khrushchev ultimately gave Kennedy two options: Accept as long as there were US missiles in Turkey, there would be Soviet Missiles in Cuba or publicly decalre it's intention never to invade Cuba, and then the USSr would withdraw. Kennedy chose the latter. Although only two months in duration, it is considered the very closest the world came to nuclear devastation.

1983 Nato Command begin a series of tests (called Able Archer) simulating a Def Con 1 Nuclear Situation, inolving radio signals and codes. The Soviets picked up on these interpreted this as cover for a real plan of attack, and subsequently went on high alert. The Soviets then began to manoeuvre units in the DDR and Poland. This doomsday clock was a about 3 minutes to midnight at this point.

EDIT My apologies for not stating clealy some issues and leaving out some details, as flagged in the comments.

During the 1983 Soviet nuclear false alarm incident, the situation got extremely close to serious danger. Stanislov Petrov was at command of the Oko nuclear early warning system. While he was serving, the warning system reported twice the launching of U.S. missiles. Petrov made the decision that these were false alarms and not legitimate attacks against the Soviet Union thus perhaps saving the two countries (and maybe the rest of the world) from nuclear war.

I feel like it needs to be stated that we weren't close to war during the cold war, we were involved in multiple very real wars as part of the cold war. It is just a matter of degree ie nuclear weapons were not used but conventional wars were waged, often via proxies. http://en.wikipedia.org/wiki/Proxy_war

Some examples. Note the list of beligerents.

The Korean war http://en.wikipedia.org/wiki/Korean_War

The second Indochina war AKA, the Vietnam war http://en.wikipedia.org/wiki/Vietnam_War

The Soviet war in Afghanistan http://en.wikipedia.org/wiki/Soviet_war_in_Afghanistan