WordPress 1.2, Apache 2.0, PHP 4.3.8, MySQL 3.23.58, FedoraCore1, https instead of http (in case that matters)
I've just experienced this on a new install (using Firefox, Mozilla or IE). I go to the login screen, enter the information and I get redirected back to the login screen. No error message or anything. If I type in an invalid password, same thing. No error message. I register as a new user, that works fine. When I try to log on with that, however, no dice. I look in the wp_user table and the new user is there. I added a print statement to wp-login.php to show me the sql statement created on line 89. I then cut and pasted that and ran it on the mysql command line and it returned the expected record (for both admin and the new user I created).
There's reference above to "that firefox thing" and "rogue 'firefox.exe'" problems but I'm not sure what that means. Can someone elaborate?