We began our intervention in South Vietnam believing that a limited war was possible, and our assistance could be geared to a limited war, but that belief was shattered over the course of only two or three years. When President Kennedy was in office, until his assassination in November, 1963, the consensus of the president and his cabinet was that the conflict in South Vietnam (it was not yet called a "war") was being won:
In the first place, we were winning the war in 1962 and 1963, up until May or so of 1963. The situation was getting progressively better. (Third Oral History with Robert F. Kennedy, April 30, 1964--see first reference link below)
Robert Kennedy goes on to point out that President Kennedy was committed to the survival of a democratic South Vietnam, but by the time of the president's assassination, the U. S. still believed a massive involvement of U. S. troops was unnecessary. At the time of Kennedy's assassination, for example, the US had only 16,300 US troops in South Vietnam, most of them advisers and trainers of the South Vietnamese Army and Air Force. Many U. S. troops accompanied the South Vietnamese on missions, and there were casualties, but not on the scale we experienced in the mid-to-late 1960s. In 1963, 122 US servicemen died in South Vietnam. Just before Kennedy's assassination, there were actually plans to pull all U. S. advisers out of Vietnam--a testament to our conviction that the South Vietnamese could prevail.
After Kennedy's assassination in November, 1963, South Vietnam's leadership, which was incredibly corrupt and ineffective, began to lose control of the struggle, and it was clear to President Johnson and his cabinet that, to save South Vietnam, U. S. combat troops, not just advisers and trainers, would have to enter the conflict. From 1963 to 1965, our troop strength went from 16,300 to 184,300, a massive increase in numbers of regular U. S. troops. Johnson firmly believed by 1964 that South Vietnam would lose the war without our increasingly active help:
By August, 1964, the Johnson Administration believed that escalation of the U.S. presence in Vietnam was the only solution. The post-Diem South proved no more stable than it had been before his ouster, and South Vietnamese troops were generally ineffective. (https://history.state.gov/milestones/1961-1968/gulf-of-tonkin)
Clearly, the US concluded that the limited war was lost and a much larger conflict, requiring massive US intervention, was open us.
Later in 1964, the U. S. Navy reported that two destroyers were fired upon by North Vietnamese patrol boats. Although there is still much controversy surrounding this incident, this led to the Gulf of Tonkin Resolution, passed nearly unanimously by both Houses of Congress. The resolution gave President Johnson the authority to defend U. S. troops in South Vietnam. From that point, Johnson began the escalation of the war--including the bombing of North Vietnam--that saw the rise of U. S. troop strength go to 536,100 in 1968. After President Nixon took office in 1968, he oversaw a gradual reduction of U. S. forces to 24,000 at the end of 1972.
It is reasonable to believe that Kennedy viewed the conflict in South Vietnam as winnable by the South Vietnamese, but with substantial assistance from the U. S., but his view, of course, ended on November 3, 1963. Just about that time, the U. S. began to understand that the conflict was not going to be won through a limited war strategy, and Johnson's view of the war was dictated by a completely different set of circumstances than Kennedy faced.