I'll just quote the second paragraph of the Wikipedia article on it:
"In 1961, the US government put Jupiter nuclear missiles in Italy and Turkey. It had also trained a paramilitary force of Cuban exiles, which the CIA led in an attempt to invade Cuba and overthrow the Cuban government. Starting in November of that year, the US government engaged in a campaign of terrorism and sabotage in Cuba, referred to as the Cuban Project, which continued throughout the first half of the 1960s."
I'd recommend The Jakarta Method for some insight into the legitimacy of the concerns of regime change.
Could those missiles be (late) reactions to Soviet activities in the areas, such as keeping their most combat-ready forces just across the Ljubljana Gap and the very recent territorial claims for Eastern Anatolia?
"In 1961, the US government put Jupiter nuclear missiles in Italy and Turkey. It had also trained a paramilitary force of Cuban exiles, which the CIA led in an attempt to invade Cuba and overthrow the Cuban government. Starting in November of that year, the US government engaged in a campaign of terrorism and sabotage in Cuba, referred to as the Cuban Project, which continued throughout the first half of the 1960s."
I'd recommend The Jakarta Method for some insight into the legitimacy of the concerns of regime change.