Humphrey Bogart received facial injuries whilst serving as a USN military policeman, or so I read in a biography. I forget the details, however these injuries resulted in his "dribbling" over the end of a cigarette. (I think his upper lip was split below the nose.) When, later, he was involved in movies, he hid this by cupping the cigarette in his hand.

I don't know the meaning of Bogart as used in the song, but I've always assumed it referred to the dribbling, not to refusing to pass the joint along.

AndyG