I found an interview with "The Fraternity of Man" bandmember
Lawrence "Stash" Wagner in "It's Psychedelic Baby" magazine that confirms my theory. Wagner said he "got down on my knees and begged" their ABC
record label to put their song "Don't Bogart That Joint" on a single,
agreeing to change the title to "Don't Bogart Me." When Peter Fonda put the song on the Easy Rider soundtrack, "Bogart" became a classic, later recorded by Little Feat and others.
On the origin on the song, Wagner said, "The band was smoking some pot in our rehearsal house up in Laurel Canyon, when Elliot [Ingber, the band's guitarist] turned to me and said, 'Hey man, don’t bogart that thing.' Elliot was always coming up with hypsterisms from the 1950’s and I loved adopting them. I asked him, what does ‘bogart’ mean? He said, 'You know, like Humphrey Bogart always had a cigarette in his hand or hanging from his lips when talking. Well, you were hanging onto that joint while your lips were flapping.' I said, 'Cool, we should write a song using Bogart.'" Three minutes later, the band had written the song.
Others have theorized that Bogart's selfish character in Treasure of Sierra Madre, his breakthrough role, is the origin of the term "bogart." According to Lee Server in the Robert Mitchum biography, Baby I Don't Care, the film's director John Houston delighted Mitchum with tales of pot smoking during the filming of Sierra Madre. In her memoir, Houston's daughter Anjelica said of the classic 1948 film, shot in Mexico, "they were all smoking grass down there, high as a clouds for most of the picture." Marlon Brando said in his book Songs My Mother Taught Me that John "did a lot of heavy pot smoking" while filming Reflections In a Golden Eye, and got Brando high before filming a scene. But what we really want to know is, did Bogart bogart the joint?
Phish did a barbershop quartet version of the song at a 2011 concert: