Do you think the plot in the book Red Storm Rising by Tom Clancy was possible?

Maybe.
I would imagine that the leaders would have been very reluctant to fire those nukes. Who knows... even if America launched or the USSR launched, the reciever might not fire... so that even if they do lose the war, at least there will be people left on earth. And if there won't be, it won't matter whether or not one side launches it or not.
Actually once in the USSR there was a time when reflections from the covers of US missile silos made it look like there was a nuclear launch in the US. The Soviet officer did not believe that the Americans launched nukes so he did not report an American missile launch.
First he was hailed for saving the world from disaster then he was villified for not carrying out his orders.
Think of that, the world could have ended because of a computer glitch.
But even THEN where the Soviet officer got the "US Launch" warning from his instruments, he did not relay it so that the Soviet Launch sequence would initiate.
So .... possible.

Red Storm Rising by Tom Clancy you say? I've read the book so many times that I sometimes believe in it - the first chapters of the book presents high politics and cover up stories in a brilliant way and I do believe we some of them to a large extent today - speaking in general terms.