How do you connect to the internet? I recall you saying it was a cellular data modem some time ago. Is that still the case?
I wonder if you have strange IP issues. Try visiting http://www.whatismyip.com/ Note the number it gives. Visit a few times over the span of some minutes. Does that number ever change? I don't need to know specific numbers, just if they are all identical or change as you visit that site multiple times over some minutes.
Have you tried to clear all your cookies and other temporary internet files? You may wish to do that, restart your computer, and try logging back in. If you're not sure how, let me know what browser you use (name and version number) and I'll find out how for you.
OK, it seems to be worsening. With an complete OS re-installation I would have expected it to be resolved. An attempt to change password resulted in the same thing. Instant logout (minus the logout applet). Though unlikely, is it possible my account may have been compromised?
If a full OS reinstall didn't fix it, it's upstream of the computer. Do you have a router connected to the computer? If not, it's your ISP doing something funny. That was why I asked if your IP address changed at all while browsing.
Is this a computer you can move somewhere else and use a different connection to access the internet? For example, if it's a laptop, could you take it to a library or a friend's house? If it works there, it's your connection. If it doesn't it's still your computer. Either way, we can then keep trying to diagnose.
Four different browsers 😀
As of today (maybe yesterday), it SEEMS to have mysteriously stopped, which is rather odd since there have been no further updates.