While with the team, Stockman won a silver medal at the 2004 Summer Olympics, and he was also a participant in the 2006 World Baseball Classic.

Originally signed by the Arizona Diamondbacks organization, Stockman was a Texas League All-Star in 2003, while working as a starting pitcher for the El Paso Diablos.

He signed with the Braves as a six-year minor league free agent in 2006, and earned a midseason promotion to the majors.

On 16 May 2008 Stockman was called up to the major leagues and pitched 1 scoreless inning of relief against Oakland on 17 May, which included his first two career strikeouts (which were Mark Loretta and Josh Beckett.)

He finished his Australian Baseball League career that season earning a 2.45 ERA in 10 games, with a miserly .067 batting average against him.