Your cookie isn't being cleared, then. Try clearing your browser of its cache and cookies and seeing if the "remaining logged in" stays. If it does then you have some sort of cookie problem we can attempt to diagnose.
If you use Firefox, do the following:
-Go to "Tools"
-Select "Options"
-Click the "Privacy" tab
-Look for "Private Data"
-Click "Clear Now"
-Make sure "Cache", "Cookies" and "Authenticated Sessions" are checked. Check nothing else unless you want to delete your history or saved passwords as well.
-Click "Clear Private Data Now"
-Close your browser and restart your computer.
If you use Internet Explorer 7, do the following:
-Go to "Tools"
-Click "Internet Options"
-On the "General" tab, look for "Browsing History"
-Click "Delete" near it
-Next to "Temporary Internet Files" click "Delete files"
-Next to "Cookies" click "Delete cookies"
-Close your browser and restart your computer
If you use another browser let me know and I'll write up a procedure for it.