This is a subjective question. Your opinion is just as valid as anyone else's. I would start by listing reasons why you feel that the Cold War raged throughout the 50's and 60's. Then you, or we, can work through who you believe was more responsible. What level class is this for?
There was a shift from a real military focus to one that was more ideological and cultural.
In the 50s, there was actual physical war: Korea and Germany had just been divided. The possibility of nuclear warfare had just been realized. Beyond physical intrusion, blacklisting had torn apart the lives of many professionals.
In the 60s, there was very little warfare. The space race was ideological. The Cuban Missel Crises was just a scare. Plus, you have American youth becoming highly liberalized. It was no longer hip to be a hetero-normative man "making the world safe for democracy." The Soviet Union saw such cultural highlights as the popularization of filmmakers Andrei Tarkovsky and Sergei Parajnov.
As for the failure to achieve peace...that is a loaded and subjective question. I'd say it's possible to only look at the US and USSR. The Cold War was highly globalized.
