An analysis of the last 10 seasons by MLB.com shows that there have been 2,137 extra-inning regular season games. In that span, Atlanta payed more extra-inning games (168) than any other team.
Noted is that the Braves hold the mark for the single season with the most extra-inning games (2011), when they played 26 extra-inning games.
So far this season, the division-leading Braves have played six extra-inning games - five were losses.
Also worth noting, teams who lead the division on May 1 most often win the division. The Braves took over the division lead from the New York Mets on May 2.