Officially VS 2008 is only supported against TFS 2013 (and Visual Studio Online) by using the MSSCCI. However, in practice it still works fine without MSSCCI, MS just no longer tests this particular scenario, so it's not guaranteed to continue working.
Make sure you have the appropriate updates installed, you need:
Order of installation is very important. Refer to this article.
Moreover, if you are still having issues connecting to TFS after successfully installing above mentioned updates, consider clearing your IE browsing history (all checkboxes) and retry.