The Cuban Missile Crisis was a conflict between the United States and Cuba in 1962, where the Soviet Union secretly built missiles on Cuba. It almost escalated into a nuclear war, and is generally thought of as one of the highest points of tension of the Cold War. The Norwegian rocket incident, however, was even worse. It took place in 1995, a few years after the Cold War had officially ended.
The incident started when a team of Norwegian and American scientists launched a sounding rocket, a rocket made to take measurements and perform scientific experiments. The rocket, which may have gone in an improper direction, was intended to study the aurora borealis in the north, but was caught on Russian radar. Not knowing exactly what it was, Russian President Boris Yeltsin had to decide whether or not to send a barrage of missiles to the United States.
Though he was supposed to make the call to activate the missiles immediately, he hesitated, and it was soon discovered that the rocket was heading away from Russia. That may have been the closest to a nuclear conflict this country has ever come to. You can read more about it at the source