Cuban missile crisis
The Cuban Missile Crisis was when Cuba and the Soviet Union attempted to put missiles on Cuba pointing towards the US. This was the first real time that US was truly worried about a nuclear attack from USSR.
The US were extremely worried that the USSR and Cuba would actually successfully launch an attack on the US. Also it frightened them because now the USSR had a close ally that would make it even easier to get to the US quicker. Kennedy agreed to remove all missiles from Southern Italy. Then Castro and Krushchev would remove the missiles from Cuba.
The USSR felt strongly about having missiles in Cuba. This gave them an advantage as to the MAD. Making it easy for them to fire near by. They did get a missile removed from a close country because of negotiations from the US, but it was an overall tie for them.
In the war this was seen as a time in which the bomb threats were becoming close to realty. Making the US and USSR anxious on who would be the first to fire.