At work, we can log into https://openid.stackexchange.com/account/login but cannot log into any other SO site with Chrome or Firefox. All scripts and cookies, including third-party cookies, are allowed.
Some notes and things I've found:
After clicking "login" using Chrome or Firefox, I get the message "Adblock interferes with login, please disable it"
I created a completely new Firefox profile with no add-ons and everything allowed to get the above results.
Thanks to the quick feedback comment, I learned my Chrome browser passes all the tests on the self-help and diagnosis webpage linked kindly by @Arjan: https://meta.stackoverflow.com/network-login-help
My Firefox browser also passes the above self-help diagnosis webpage
In Internet Explorer 8 (IE8), I'm able to login.
I can successfully log into https://openid.stackexchange.com/account/login using Internet Explorer, Chrome, or Firefox. This login session can't be used to access other Stack Exchange sites.
We have a proxy at work, though I have no access to its settings.
At home I'm able to log into any/all sites without issue.
It seems there's something different in the way Stack Overflow sites handle the Chrome and Firefox engines, otherwise we wouldn't be able to login with Internet Explorer.
It also seems the openid.stackexchange.com site handles logins differently than the other SO sites.
Though this problem is unique to our work environment, there's enough things in the way the SO back-end is set up to persuade me to post this. If others are not experiencing it now, they likely will in the future.
If there is something I can do to gather more information, like doing a Firebug or Fiddler capture, please let me know - though I'll need to be told what to configure and how, and what to look for.
I look forward to either hearing something I missed doing/not doing, or learning something neat about the SO login process that allows me to login using Chrome or Firefox, or learning there's a configuration setting on the Stack Exchange side of the house that could be tweaked to allow me (and others?) to log in using Chrome and Firefox again.