Bream has a brother known as Sid Bream (Sidney Eugene Bream) who is an American former professional baseball first baseman. Sid played in Major League Baseball (MLB) for the Los Angeles Dodgers, Pittsburgh Pirates, Atlanta Braves, including Houston Astros between the years 1983 and 1994.

Sister, Theresa, and brother, Stan, were pretty good athletes as well as cousins Doug, Dwight, and David, who each stood out on the hardwood and the diamond. Bream is best known by his game-winning run scored in the 1992 National League Championship Series that sent the Braves to the World Series.