We were incredibly close to nuclear war during the cold war on at least a couple of ccasions
1) During the Cuban missile crisis, Soviet subs were trying to sneak their way to Cuba but the US blockaded the island and searched for these vessels and once they found the subs, they tried to get them to surface
The Americans decided to ratchet up the pressure, and dropped warning grenades into the sea. Inside the sub, the Soviet submariners thought they were under attack.
Valentin Savitsky, the captain of B59, was convinced the nuclear war had already started. He demanded that the submariners launch their (nuclear) torpedo to save some of Russia's pride.
The programme on Channel 5 revealed how in any normal circumstances Savitsky's orders would have been followed, and World War Three would have been unleashed. Savitsky hadn't counted on Arkhipov. As commander of the fleet, Arkhipov had the last veto. And although his men were against him, he insisted that they must not fire - and instead surrender.
It was a humiliating move - but one that saved the world. The Soviet submariners were forced to return to their native Russia, where they were given the opposite of a hero's welcome.
http://www.dailymail.co.uk/news/art...verted-WWIII-height-Cuban-Missile-Crisis.html
(there is a channel 5 documentary but it's blocked on Youtube so i can't link it here)
2) Russian early warning systems detected an American launch of ICBMs and luckily the commanding officer decided to ignore these warnings and didn't launch and they turned out to be the sun glinting off clouds which the computers interpreted as a launch
https://en.wikipedia.org/wiki/1983_Soviet_nuclear_false_alarm_incident
(there is a channel 4 documentary but it's also blocked so i can't link it here either)
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If the nukes didn't exist, would they have thought twice about starting a war?
The cold war didn't start just because of the existance of nukes, but it could be argued that they prevented it from spiraling into full blown conflict - is that worth the cost of the weapons systems (collateral damage prevented, lives saved, etc...)
World politics change (see the collapse of the USSR, but you don't know whats around the corner and it could easily revert to a similar stand off again between 2 major countries and a new cold war start)
You'll find this fascinating as well...
http://www.theguardian.com/commentisfree/2012/oct/15/cuban-missile-crisis-russian-roulette
Accounts from the USAF with regards to the lack of control that US commanders had over their forces in the air during the crisis. In their case, there was no veto, and October 26th was a very dodgy day.