The son of Dr. J. (James) Dougal Bissell (1864-1935) and Helen (nee Nutting) Bissell (1878-1977), Bissell's had at least one sibling, a brother, Dr. Bradford Bissell (1907-1949), Whitner Nutting Bissell was an avid yachtsman and fencer in his youth.
He attended the University of North Carolina in Chapel Hill and while there began a serious pursuit of acting with the Carolina Playmakers. After a stint on the Broadway stage, Bissell began a career as one of the most recognizable character actors in the history of film and television.