OK this is going to sound absolutely ridiculous. I am completely aware of this, but I can only report what seems to be the case.
Connection Doctor implied it was a password issue ... which wouldn't make sense because it had just stopped working randomly.
I have two 1and1 email accounts set up on my computer.
Somehow, the passwords for the two seem to have switched. In other words, the password I used to use for Account A now works for Account B and the password for Account B now works for Account A.
I know that sounds absolutely ludicrous, but I have certainly not EVER manually changed the passwords, and I know very confidently what the passwords were for each account. But I just did something crazy and tried switching them and it seems to be working now.
This isn't some weird 1and1 issue because the two aren't even part of the same 1and1 account. They are registered COMPLETELY separately.
As further evidence of this, I had the Account A password saved for the webmail log in page, and this no longer worked just now (despite the fact that it worked minutes ago, when I tested for a post earlier here), but I tried the password for the Account B and it did. I have absolutely no idea how this happened.
And I'm really not that bothered about working it out. My imagination suggests some crazy **** could've happened with iCloud account syncing, but even that sounds totally totally ridiculous.
Anyway, it's sorted, that's the main thing ... thanks for your help again ...