Personally I would have a hard time suspending disbelief if a story set in 2200 didn't have super-human AI and all diseases cured. Similarly there should be no pollution or over-population, at least in first world countries.
For humanity to avoid achieving those things, there would have to be some significant barriers to progress that we don't know about yet, or a fairly disastrous collapse, which would limit the possible stories that could be told.
I suppose that setting a story thousands of years into the future allows there to have been multiple collapses and renaissances, but I'm not convinced that such a path is as likely as a permanent collapse or reaching some sort of uninterruptible paradise state.
