Well, about three analysts came and went before I got a dude I'll call Reggie.
The first thing Reggie tells me to do is input my password, which I do (just 'cause) and then explain to him that it isn't working. He goes 'huh'. Then he asks about my input/output mail servers, tells me one of them is wrong, and later reverses that opinion when I tell him that nope-- my computer reads the server he was trying to tell me to get on as 'unverifiable'.
Now, this entire setup has been working for me for years. Literally years. I know that my server settings and ports and such are all correct. According to Reggie, however, they are all wrong. So I humor him, write all my settings down (it's worth noting that Reggie is impatient and will send the same message three times if you're silent for a minute), and change them. Big surprise, it somehow manages to work even less. So I change the settings back.
Then, he finally suggests what I've been wanting since I went to the website-- for my password to be changed. He gives me a new password, sends me to log in to the site with it (which works), and then tells me to try to check my email. Doesn't work.
“Try it again,” says he.
“Still doesn't work,” says I.
“Huh,” says he. The (_______) password isn't working?”
Me: No, the (_______) password is not working.
Reggie: ...huh. And you're sure it's not the settings?
Me: Pretty darn.
Reggie: Well... it must be on your end. Try uninstalling.”
Me: Uh, okay. Thanks for your help. *kind of uncertain, because at this point I don't know if he knows what he's doing and it sounds like a cop-out*
Reggie: It has been a pleasure assisting you with this problem!"
tl;dr: email stops working, I contact tech support, and tech support tells me that all my settings are wrong. When I prove that the settings are right, they declare that the entire problem must be on my end. Problem still not resolved.