He grew up in Rye, New York with his mother Alma Mary Court (a variation of whose maiden name he would later use professionally), his father, Joseph Parker Cox, Sr., who suffered from multiple sclerosis, three younger sisters and an older brother. His Irish American father was a bandleader, pianist, and merchant, his mother, a reporter of Irish Catholic and French Canadian descent who took over the family clothing business when her husband took ill. Bud Cort graduated from the Roman Catholic Iona Preparatory School in New Rochelle in 1966.

He entered New York University to study design in 1967, but dropped out in 1969 to pursue acting. His first role was as an extra in the film version of Up the Down Staircase, and he then went on to study acting with Stella Adler and do television commercials, as well as stand-up comedy in New York City. He was discovered in a revue by Robert Altman and cast in two Robert Altman movies, M*A*S*H and given the leading role in Brewster McCloud.

Cort next went on to his most famous role, as the suicide-obsessed Harold, in Harold and Maude. The film was not a hit when it came out, but later gained international cult status. Cort feared being typecast after playing Harold; he initially was offered the role of Billy Bibbit in One Flew Over the Cuckoo's Nest, but he wanted the role of McMurphy instead, a role which ended up being played by Jack Nicholson. Because director Milos Forman would not give it to him, (before Cort knew that Nicholson had been offered the lead part) and as he did not want to be stereotyped in another film as "a crazy," he turned down the role. Instead he turned to a career in theater and night clubs.

In 1979, Cort was in a serious car accident on the Hollywood Freeway. On the way home from a Frank Sinatra concert, he rear-ended a car which was abandoned in the road. His injuries included a fractured skull, severe cuts to his face, the loss of several teeth, and a broken arm and leg; he had reconstructive plastic surgery, and subsequently lost a court case against the owner of the abandoned car. Having spent months in recuperation, he was out of the public eye and subsequently suffered career difficulties, with his work being relegated to small character actor roles in primarily independent films. His last leading role, which he also co-wrote and directed, Ted and Venus, garnered unfavorable reviews. He also worked during this time as a radio and voiceovers artist.

During the 1980s he was a familiar face on TV and appeared in such films as the remake of Invaders From Mars. He continues to act, and recently appeared in Kevin Smith's Dogma, Pollock, and the latest film by director Wes Anderson, The Life Aquatic With Steve Zissou. His most recent TV appearance was in the season finale of Arrested Development in February 2006.