This morning Mike Butcher wrote a great review of TwitterLeague on TechCrunch UK. Just 10 minutes after the article had gone live 20 people signed up. About 8 hours later the site experienced some problems when processing logins. This problem is caused by the Twitter API limit.
This is because whenever you login the site the application goes to Twitter and basically asks “are the username and password correct?”. The site the gets a response from the Twitter API of “true” or “false”. But when the API limit is reached there is no longer a true or false returned, jsut a line of text saying “The API limit has been reached” which means users cannot be authenticated and logged in.
I have applied for TwitterLeague to be whitelisted and so until then there will be login problems. 
I really am as frustrated as you. I didn’t expect to reach the limit. It is just a painful wait until Twitter approve the app for whitelisting.
Thanks for hanging in there!
Ollie