One Solution
So I had the same problem many of you had: after a year or so without error, airport began saying that I was connected to a network, even though I wasn't getting any access to the internet.
After a long troubleshoot with an Apple guy, I have found at least one fix to this problem. This fix will be for those who are confident their problem isn't hardware.
Solution: First of all, Snow Leopard caches all the networks you've ever accessed (unless you've changed that particular option). Now for some reason, when you have accessed a fairly large number of networks, the bug comes up where you continue to "connect" to networks without actually connecting to the internet.
In order to remedy this issue you need to delete your cache of wireless networks.
To do this, first go to the Network menu in System Preferences. Highlight the Airport device in the box to the left for network devices and then go to advanced options. In this options menu make sure you are on the "Airport" tab. Once you are there, select all the networks that are listed in the Preferred Networks box and click the minus button to delete the cache of networks. With luck, this should free up your wireless.
If your internet isn't working immediately, I would recommend simply restarting your modem and router if you have access to them.