Bunce's career began in the early 1930s with small roles in several films and guest appearances on several radio programs.

Bunce played New York State Governor Al Smith in "Sunrise at Campobello" about President Franklin Delano Roosevelt's early battle with polio and return to public life during the 1924 Democratic convention where Roosevelt walked ten steps to the podium and stood for 45 minutes to nominate the Governor for president.

[citation needed] Bunce broke into television in 1950 with a guest star appearance of the ABC anthology series The Clock.

Bunce's "Albert Arbuckle" character also moved to television starting with appearances on The Kate Smith Hour beginning in 1952.

It moved to CBS on June 20, 1955, and stayed there until September 26, 1955, serving as a summer replacement for December Bride and ended its television life on ABC where it aired from October 14, 1955 – July 6, 1956.

In his later career, Bunce appeared on episodes of Perry Mason,[12] The United States Steel Hour, The Patty Duke Show and The Nurses.