Remember, these are films that DIDN'T win best picture.
Mine is "Seven Days in May" 1964, starred Burt Lancaster, Kirk Douglas, Eva Gardner and Frederic March (as the President). The screenplay was written by Rod Serling.
The movie was about a planned coup d'etat by the Joint Chiefs of Staff, because of thier opposition to an arms treaty that had just been signed by the President. Extremely tense and well written, with excellent acting by supurb cast.