This is for the era prior to 1970, when ballistic missiles and early warning systems were not as developed.
I've done quite a bit of research on Air Defense Command, and it is rather striking how poorly it performed in exercises. The Semi-Automatic Ground Environment System would have helped to improve air defense performance, but it was only fully operational for a few years before ballistic missiles became the main threat. It was also apparently quite vulnerable to even rudimentary jamming.
To make matters worse, Strategic Air Command wasn't actually allowed to possess nuclear weapons during its early years. There was a fear of military control of the nuclear arsenal, and to ensure civilian control weapons were not only legally owned but also possessed by the Atomic Energy Commission and stored at a single site. Under these procedures, if the President ordered a nuclear strike Strategic Air Command would have to fly bombers to the Atomic Energy Commission storage site to actually retrieve the weapons, a process which could take some time. I think the mate/demate equipment for nuclear weapons only existed at that facility too.
Later procedures allowed Strategic Air Command to possess nuclear weapons as deemed necessary, and eventually the Atomic Energy Commission simply shipped nuclear weapons out to military bases without ever going into storage. Still, in the early period all or a significant portion of the nuclear arsenal was held at Atomic Energy Commission facilities. The weapons program was also dependent on a handful of sites for production of weapons material, and there are only so many Strategic Air Command bases.
Curtis LeMay moved Strategic Air Command to ground alert status shortly after taking command, but Air Defense Command studies found the risk of a surprise atomic attack to be relatively high.
So, was there ever a period in which the United States or Soviets could have executed a surprise atomic attack against the other? Could a flight of Soviet Tu-4 bombers on a one-way flight have managed to destroy the American nuclear deterrent during the early Cold War, leading to a prolonged conventional conflict favoring the Soviets? Could SAC have infiltrated the Soviet Union in 1962 and destroyed most of their nuclear force on the ground in a first strike?