Anyway, this morning I was trying to connect to open a project on a machine I hadn't used before. I did this by going to the TFS web portal and clicking on the "Open new instance of Visual Studio" link. Unfortunately, rather than seeing the project open in Visual Studio I was presented with this:
TF30063: You are not authorized to access https://XXXXXXX.visualstudio.com/defaultcollection.
After much investigation it seems that while I was viewing TFS in IE with one set of credentials, the embedded browser within Visual Studio was previously logged in to a different TFS server with a different set of credentials. As the set of credentials cached/remembered in VS weren't valid for the new server I got this error.
Open the web browser window in Visual Studio.
Log out of the old server and log in to the new one with the correct credentials.
The "Open new instance of Visual Studio" link now works.
I hope this knowledge being documented here helps someone else avoid this issue or solve it faster than me if they do hit it and then find this post.