Likewise, instances have their own system of servers, which is why you can sometimes be in an instance when the world server "outside" will go down, and if you leave that instance you'll get disconnected. And Instance servers can be overloaded as well. This error message is likely a way for Blizzard to keep from crashing the instance servers -- if too many instances of a dungeon have already been created, new players trying to get in are not allowed.
However, as a few people say in the comments of what9000's post, there are some client-side fixes as well. Sometimes, the Instance server will lock you out just for a single instance's creation, so resetting the instance on your side (by leaving a group and rejoining) will sometimes clear things up and let you in. It's not impossible that Blizzard's servers are overloaded (we should see a nice bump of new players this weekend, after the holiday), but if you're seeing this message constantly, even during slower times, something is likely wrong.