What failures occurred during John F. Kennedy's presidency?
I would argue that John F. Kennedy's major failures were in the area of foreign policy. They included the Bay of Pigs invasion and the escalation of US involvement in Vietnam.
The Bay of Pigs invasion was a complete fiasco. The US supported the invaders enough to make the invasion happen, but not enough to give it any chance of success. Allowing the invasion to happen showed poor judgement and was a clear failure.
It was also under JFK that the US got more deeply involved in Vietnam. On his watch, the US supported Ngo Dinh Diem even as he oppressed many of his people. Then the US allowed him to be overthrown and (probably without US approval) assassinated. This led to a US commitment to back a series of governments that had very little credibility with their own people.
These two things constitute major failures of the JFK presidency.
The Bay of Pigs invasion was probably the most conspicuous failure on Kennedy's part. In an ill fated attempt to undermine the Communist regime of Fidel Castro, CIA trained Cuban exiles invaded Cuba shortly after Kennedy's inauguration, but were defeated rather quickly and easily. In an attempt to camouflage the United States role in the invasion, Kennedy declined to provide air cover for the invading exiles. This was a smaller scale battle that was part of a larger competition between the United States and the Soviet Union for nuclear supremacy and control of nations that might be easily influenced one way or the other.