Bancroft was a three-sport star at Fresno High School and enrolled at the University of California in 1925. He played on the Cal Freshmen team in his first year as a tight end. He also filled that position on the varsity in 1926 and 1927, but was moved to tackle in 1928. During the "Big Game" with Stanford, he intercepted a pass and ran seventy-eight yards for a touchdown in a 13-13 tie. He was one of Admiral Nimitz's favorite pilots during World War II. Following his discharge from the Air Corps, he became a Pan American pilot.