I'm having this same problem, but only at certain times. The first time I try to login during a new browser session, the login box hangs with the "checking credentials" message displaying. I can exit the modal box and login to the site using the standard WP login page and I get into the site. Subsequent login attempts with the modal dialog box are successful. If I close the browser, then start a new session, I'm back to the hanging box for the initial login to the site. This happens for my admin role, and for several "dummy" subscribers that I use for testing.
I'm using Windows XP SP3, WordPress 3.5.2, and I'm getting the same experience using both Firefox ver. 22 and IE ver. 8. I'm using a plugin named "Anti-Captcha" which I disabled, but to no effect.
The solution another user found several weeks ago doesn't apply to me as I don't have that code in the php file that he referenced.
Full disclosure: while I'm reasonably computer literate, I'm relatively new to blogging. I've done no customization that requires writing code at this point.