Fail-Safe opened in October, 1964, about ten months after Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb. Both movies were produced by Columbia, although Fail-Safe was originally produced by United Artists (see below). When Stanley Kubrick found out that Columbia was attached to both productions, he insisted that his movie be released first, as the knew the second movie released would be unfairly compared to the first one and not do as well at the box office. That is what happened to Fail-Safe, although interest has been high in recent years due, in part, to the excellent performance of Henry Fonda and direction by Sidney Lumet.
Novelists Eugene Burdick and Harvey Wheeler had been sued for plagiarism by Peter George, whose novel, "Red Alert", was the basis for Strangelove. That case was settled out of court and part of the settlement was that Columbia was given the rights to "Fail-Safe" (the book) and hence the film. However, in 2000 Harvey Weeler stated in this article at http://www.geocities.com/dcjarviks/Idler/vIIn32.html that Fail-Safe was based on a book he wrote in 1956 called Abraham '58 (published as Abraham '59), before Red Alert was published. Furthermore, Wheeler goes on to say that Peter George had been "fiction editor at more than one of the U.S. magazines I'd sent Abraham 58"! The reason that Wheeler and Burdick capitulated and gave Columbia the rights to Fail-Safe was that the producer, Max E. Youngstein, was out of money and that this was the only way of getting the film made.
During the 1950s, tension between America and Russia was high, with many believing nuclear war was either inevitable or very likely. It is probable that both "Red Alert" and "Fail-Safe" were born out of this fear, with the film industry cashing in, possibly spurred on by the October 1962 Cuban Missile Crisis.