Actually, there was an incident, I believe in the 1980's, that was probably the closest we came to missile launch. During NATO exercises, a Soviet early warning system was somehow activated and indicated we were attacking. The officer in charge had standing orders in such a case to immediately activate nuclear launch but thought it was strange that we'd be attacking with only one missile, so he held off and was able to verify the error. I seem to recall his reward for preventing nuclear catastrophe was a courts martial and punishment for disobeying orders. I shudder to think about the outcome if a more "obedient" officer had been in charge that night.
The most dangerous moment of the arm's race is the NATO military exercises that took place from 1979-1989. The reason why it was dangerous is because the conflict was expressed through military coalitions, espionage and nuclear arms races. Visit http://en.wikipedia.org/wiki/Arms_race for more information.
Most would agree the Cuban Missile Crisis was the most dangerous moment during the Cold War, the History Channel did a special a few years ago and showed how easily it could have escalated into war, the Cubans were shooting at American recon planes and American ships blockading Cuba were firing warning shots near Soviet ships carrying missiles, both sides managed to ease back and solve the crisis but we came frighteningly close to war!