The hockey teams left. (Flames, then the Thrashers)
I don't know if this is it or not, but life was always pretty good in Atlanta -- at least it used to be. Back in the early '80s it was voted "the most livable" city in America. Such a place can breed a lot of fair-weather fans compared with tougher places to live like Chicago or Detroit or even St. Louis, where sports is a great diversion from the stresses of everyday life. Another reason could be that pro sports came late to Atlanta... When the Braves and Falcons arrived, Atlanta was no bigger than Durham, NC is today. At one time, all you had were the Yellow Jackets and the Atlanta Crackers. Then within the space of a few years, you get the Braves, the Falcons, the Hawks and the Flames. And there was also the Chiefs. The market suddenly gets divided among many different teams. Now add to the mix a bunch of new transplants to Atlanta over the following decades, many of them bringing their old loyalties with them, and the home team never really gets a clear home field advantage.
For years there was also a lot of negative bias against Atlanta-based teams from the sports press, partly because Atlanta was an up-and-coming city and wasn't supposed to compete against the New Yorks or the Bostons or the Chicagos of the sports world. These cities were supposed to produce champions, not an upstart city like Atlanta. I think that inferiority complex may have filtered down to the players on some level. Combine that with fair-weather fandom, and that's a possible recipe for mediocrity and post-season collapses.