I do work for two different companies and have a Microsoft account/identity for each. I created a Team Foundation Service account (http://gallerygorilla.visualstudio.com) for the first company under my first Microsoft account and successfully connected via Visual Studio and created and checked in a simple test project. I then created a Team Foundation Service account (http://metrix411.visualstudio.com) for the second company under my other Microsoft account and was again able to create a simple test project and check it in.
However, when I now go back and attempt to connect to the first TFS account, I get a TF31003 error. The "Retry" link does not work. Neither does the "Sign In" link. I have since tried using the "Open new instance of Visual Studio" link from the browser based viewer of the first account but still get the same error.
The first account is still good as I have been able to connect to it from a virtual machine. But I have had absolutely no luck on my main dev machine. It seems as if VS 2012 has fixated attached itself to my second Microsoft account and there is no way to get it to "forget" the second account and use the first account. I would have expected that when clicking the "Sign In" link on the "Connect to Team Foundation Server" dialog, I would be presented with fields to enter the credentials for my first Microsoft account. However, it appears to be simple trying to connect to the first TFS account using the credentials associated with the second TFS account.