I'm having the same problem. Sometimes it keeps me logged in, sometimes it doesn't. I know there was some problem just after GL switched hardware - to be honest I assumed that once they'd fixed that, it was simply a matter of caches catching up, but it's been a while and still seems to be a problem, so I'm not sure. I'll check with my guild members and see if they're having similar problems.
I'm assuming they must have been able to at some point - I don't think you can post here unless you're logged in? Certainly wouldn't let me, anyway.
Might be interesting to find out what browser is being used ... Firefox has been a bit finicky lately - it refuses to automatically log me in to some sites now, and sometimes warns me when I try to.
This may help: