I have had exactly this problem with my POP email accounts. I have raised a bug on it with Apple, but not had any suggested fix for it.
In my case, Connection Doctor showed I was connected, but I still couldn't download email from my POP accounts. The IMAP accounts were fine.
I've gotten sick of people telling me to switch to IMAP (no offence intended) but since I was crippled without the ability to download email I took the hit and it is, indeed, 100% better than POP when accessing from multiple devices.