The same issue for years now.
What I did notice is that since I made my IMAP/POP3 a manual send and receive, which I only do a couple of times a day, no issues for 3 weeks (fingers crossed).
Sounds an interesting idea. However I often receive emails that need action, so it wouldn’t work for me.
Yesterday I posted that I couldn’t send or receive on 3 devices, I took no action and, fortunately, in the evening the mobile went ping and everything was back to normal.
It’s annoying that BT always insists that the problem rests with our equipment when it obviously doesn’t, and then shuts the door in your face by declaring the only service they support is BT Mail.
I accept that they can’t provide assistance if you have a problem with your equipment, but they do provide the service and therefore should provide assistance when their equipment that fails.
Now here is an interesting situation. Going through my diary for last year, I found that I had had this problem before and how I fixed it.
My antivirus - AVG Internet Security - that's the paid version of AVG - has a specific setting for checking emails as they are downloaded. My solution last time was to switch off that check - download some mail and then switch it back on again. It will re-check whatever you have downloaded anyway so there is no risk. What was happening is that it was downloading the email headers but something in the a/v was clashing either with Thunderbird or the BT interface such that the email body never got downloaded as well.
Tried it and it worked instantly. I have been back in touch with AVG. They previously wanted me to do a live session with an engineer to try to get to the bottom of this problem, but was not happy having them in control of my PC. So they are working on it to try to reproduce the problem themselves.
It was a suggestion from my ex. who is a network engineer for Garmin that led me to try this solution. May work with other a/v's or other email clients. Worth a try. AVG will nag you to switch the email protect back on if you forget, don't know about others, so don't forget.