Update 8/8/2013 3:31 AM - There has been a resurgence of this issue, and we highly recommend players explore the workarounds described below...
We have performed a thorough investigation of all our services, hardware, and datacenter connections since Saturday afternoon. After reviewing traceroute data from players, we have identified a broader internet issue with data routing external to our servers and networks. This morning we received word that one of our affiliated connection providers is taking steps to re-route services around certain areas of the internet that are experiencing issues. We believe this to be the root cause of the reported issues.
Their work may take some time to propagate through the internet in general. If you are able to, take advantage of the temporary workarounds to use an alternate connection method (such as VPNs, tethered cellular, or acquiring a new IP address from your ISP) - any alternate internet connection that does not happen to route through the cached network paths from your existing provider. This may allow some of you to enter the game until such time as this rerouting and propagation completes.
We appreciate the efforts by those who discovered workarounds, other reports of similar issues elsewhere on the 'net, and all of you who submitted tickets detailing your issues and locations and other requested data. (Please also keep in mind that this issue is not the same as the issue that causes some users to need to log in twice before they can connect.)