I don't know what the problem is, but the steps you have taken are on the right trail. something is blocking the cookies. With the variety of browsers you have tried, it seems unlikely that is is because of any cookie settings in them, but I would still go into cookie management and see what cookies are actually there. If there are any for those particular websites, I would delete them and try logging into the websites again...just in case. Sometimes when I have had similar problems, it was due to an extension that I use called Ghostery, so I had to make specific setting in it to prevent it from blocking them. I really doubt that it is because of some system problem, I would imagine that it is because of some software or setting that either you or someone using your computer has installed or made.
EDIT: Check the Privacy tab of Internet Options, and make sure it is set to a setting that's not too restrictive.