Kennedy is the exception to the VP rule losing in 1956 and Hillary will compare herself to him. (unless you consider that 1960 election fixed, which it may well have been)
Kennedy didn't run for VP in '56. Estes Kiefauver was Stevenson's running mate. At the '56 convention, Stevenson threw the VP nomination open in a surprise move, and Kennedy and Kiefauver were placed in nomination. Kiefauver narrowly won the nomination.
Kennedy did get a very small name recognition bump from this. But it was not at all the same thing as "running for VP." In fact, maybe he benefitted more by NOT actually being Stevenson's running mate.