The '60s did not really get going in 1960, nor end in 1969. As we know it culturally, the decade began with the JFK assassination in 1963 and ended only with Nixon's resignation 11 years later. Neither, many historians say, did the 19th century finish in 1899. Instead, it continued until WWI — and even as long as the 1930s.
The big picture: Historians call these "long decades" and "long centuries" — super useful constructs for grasping big, prolonged events outside of mere time, and seeing them as eras, zeitgeists and social phenomena.