.I also can't help but think that Dr. Strangelove and Fail-Safe would make an interesting double bill. They share a very similar plot line, but with vastly different treatments of it.
From the IMDB:
Columbia Pictures produced both this movie and Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb (1964). Director Stanley Kubrick insisted his movie be released first, and it was, in January 1964. When Fail-Safe (1964) was released, it garnered excellent reviews, but audiences found it unintentionally funny because of "Strangelove", and stayed away. Henry Fonda later said he would never have made this movie if he had seen "Strangelove" first, because he would have laughed too.
Peter George wrote the novel Red Alert upon which Dr. Strange love was based and shared writing credits with Kubrick. He also worked on the script for Fail Safe but did not receive any screen credit.
Eugene Burdick and Harvey Wheeler's later bestseller Fail-Safe so closely resembled Red Alert in its premise that George sued on the charge of plagiarism, resulting in an out-of-court settlement. Both novels would inspire very different films that would both be released in 1964.
You could make a triple bill by throwing in Nevil Shute's On the Beach (1959) which is cited as an inspiration to both.