What failures occurred during John F. Kennedy's presidency?
Kennedy's record on civil rights was something of a mixed bag. Initially, his approach to the question was piecemeal. As a Democrat, Kennedy knew that he was reliant on Southern white supremacists in Congress to advance his legislative program. He'd also need Southern votes for re-election in 1964.
Though personally sympathetic to the cause of civil rights, Kennedy tended to react to events as and when they happened, rather than show leadership on the issue. Kennedy thought that by working behind the scenes and striking deals with Southern politicians, he could promote civil rights without undermining either his Presidency or his chances of re-election.
The failure of this approach was seen in the case of James Meredith, an African-American student who attempted to enrol at the University of Mississippi. Despite reaching what he thought was an accommodation with the state's white supremacist governor, violent segregationists prevented Meredith from enrolling at the university.
Even this very public humiliation did nothing to change Kennedy's piecemeal approach to civil rights. He continued to submit watered-down proposals to Congress that failed to address key demands of civil rights campaigners, such as equal access to public facilities.
It was only after the violent attacks on civil rights marchers in Birmingham that Kennedy started to take action. He went on national TV and announced to the nation that he would send comprehensive civil rights legislation to Congress. But five months later, at the time of Kennedy's assassination, still no legislation had been passed.
Kennedy's failure on civil rights was by no means an absolute one. It's rather that he grasped the full moral import of the issue much later than he ought to have done, slowing down the pace and extent of civil rights reforms.
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.