I have the same problem. I spent three hours on the phone with Verizon. The very nice tech fixed it, but mail stopped working again. I tried all of the great suggestions here, but none of them worked. I installed Thunderbird and have the e-mail forwarded to that account. The crazy thing is that I could go to webmail and get and send Verizon e-mail with no problem. But having spent a lot of time on some Apple community forums, this seems to be a huge problem and not just for Verizon. Ugh.
Find out it's Verizon's way to DISABLE ALL 3rd party email software (Eudora, Outlook, etc.); only Verizon's system works.
Actually at this point all that I have seen posts for here do work for those using regular Verizon email. There does appear to be trouble for some users with Verizon's Yahoo interface for Outlook 2003 users.
Very few use Verizon's email client.
Eudora problem was solved by disabling apop. The solution posted by jfadden is to add the following line to Eudora.ini in the section labelled [Settings]: