Using Mozilla 0.8, I have to log in again at each step that requires a valid user name. This works fine with Netscape 4.76, and after looking at the cookies for both, I can't see any substantial difference. See also bugs 15344, 21025, and 30860, which are either for older bugzilla versions or don't have enough detail for me to decide if it's the same thing.
I am unable to get this to fail when using mozilla-0.8 from my end. Have you tried removing all Bugzilla related cookies from your mozilla cookies and try again? I have found that sometimes having more than one cookie of the same name for a site can cause problems and clearing them all out and allowing only one to be set fixes things. You can do this in mozilla by going to Tasks->Privacy and Security->Cookie Manager->View stored cookies and then just highlighting each one from bugzilla.redhat.com and removing.
You're correct; clearing all bugzilla.redhat.com cookies fixed the problem. Thanks!