Thanks for the suggestion, Slyclops, but it's not just a problem with browsing the Internet. Even if I wanted to, this company blocks external DNS servers (like Google's 8.8.8.8; can't even telnet to it).
Yes, the WiFi connection does not drop (ifconfig stays the same), but all network connectivity comes to a halt. For example, I was pinging the internal gateway (IP), and when everything stopped working (browsing, skype, etc), the ping also started timing out.
Because of this, my gut is telling me that this has something to do with changes in the APs that are not being handled gracefully by the wireless driver. The place where this happens the most is a corporate campus where load balancing on the access points takes place automatically. Normally, you just get bounced to another AP, but it could be that once the MAC address of the AP changes, the driver goes out to lunch...