The tensions between the United States and the Soviet Union culminated with the Cuban missile crisis and came very close to turning the Cold War into a nuclear one. Thankfully, it never came to that, but not thanks to those who were willing to play dirty. There were plenty of them on either sides.
Several months prior to the October Crisis, the US Department of Defense and the Joint Chiefs of Staff drafted a series of proposals that aimed at creating incidents had the appearance of Cuban attacks against the United States, giving them an excuse to retaliate and overthrow the Cuban government. It basically called for acts of terrorism to be carried out on US soil, against its own citizens. Named Operation Northwoods, it would rely on the CIA and other operatives to hijack planes, orchestrate bombings, shoot innocent people and commit violent acts of terrorism in Washington, Miami, New York and other major cities.
Afterwards, fake evidence would incriminate the Castro regime and the US would have internal and international backing to launch their war.
They even considered sabotaging the Mercury-Atlas 6 spaceflight, the mission that put the first American on Earth’s orbit. At the time, the mission was the pride of all Americans and an important leg in the space race. However, the JCS had a different plan. Their objective in the event of the MA-6 rocket exploding was to pin it on the Cubans no matter the cost.
The Kennedy administration rejected their proposal.