This is definitely an issue between Apple Mail (which, frankly, bytes anyway) and/or the keychain, but I didn't bother refreshing the keychain...I got another mail app and the problem disappeared. I get ALL my mail, with no problems and I even got Rules to work, which I never could in Apple Mail. Plus, there are a bunch of addons (I haven't tried any) that look pretty neat.
It's not a cheap solution, but I wasn't willing to wait around until Apple got their act together and fixed it...I'll be old by the time that happens.
Postbox 3 $29.95 USD ($9.95 USD for upgrade only, if you have the previous version): http://www.postbox-inc.com/
Addons: http://www.postbox-inc.com/extensions
There are plenty of other mail apps, all touted as 10 times better than Apple Mail (mainly because they work and Apple Mail doesn't). Sparrow is another one, although I never tried it, and it's only $9.95 USD: http://sparrowmailapp.com/mac.php
Both of these have free downloads (Postbox is a 30 day trial and Sparrow is ad-supported) from the developers, or you can get Sparrow in the Mac App Store. There are others, but I'm so happy with Postbox I didn't look any further.
N.B. Some odd things have been happening with Apple, IMO, since Steve Jobs died...and it hasn't been for the better. At least, not for customers. Oh, well. At least there are developers out there ready to give us stuff to make our Macs work right. LOL 😁