Focusing on the three major professional teams, there hasn't been much elite play.
While the Braves thoroughly dominated the National League throughout the 1990s and early 2000s (until the playoffs, save for the 1995 World Series title, which remains the most recent championship for the major teams in Atlanta), they've advanced to the postseason only three times in the past five years. Of those appearances, none saw the team get past the NLDS.
Though the Falcons have been consistently above average until last season's 4-12 collapse, they've won only a single playoff game in the past five years. That came in the 2012-13 divisional playoffs against the Seattle Seahawks, but the run quickly ended at the hands of the San Francisco 49ers.
The Hawks are in a similar boat. Though Al Horford and Co. have advanced to the playoffs each of the past five seasons, they've never done anything more than win a first-round matchup. That might have changed this past year, but a Game 6 lead was blown against the top-seeded Indiana Pacers and put a premature end to a potential run.
There's a reason Atlanta fans have come to expect things to go wrong at the worst possible time. Regular-season excellence is expected. Postseason success is a dream, but it's almost widely acknowledged as one of the pipe variety.
Each of the major sports is mired in mediocrity, though the collegiate teams tend to do extremely well. Georgia Tech advanced to the NCAA tournament as recently as 2010, and UGA football is always competitive, even if it seems to perennially underachieve.
The most successful team, though, would be the Dream.
Led by Angel McCoughtry, the WNBA representatives have made the playoffs each of the past five seasons and advanced to the WNBA Finals three times, losing to the Seattle Storm once and Minnesota Lynx twice.