Not sure if this fixes it or not but some have suggested on bt forum that you change your Server setting | Server name to "mail.btinternet.com". Maybe it was previously "imap.btinternet.com". I recommend a tb restart after changing this. This is just a shot in the dark so still need the logs (unless this really does fix it).
Bug 1571655 Comment 5 Edit History
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Not sure if this fixes it or not but some have suggested on bt forum that you change your Server setting | Server name to "mail.btinternet.com". Maybe it was previously "imap.btinternet.com". I recommend a tb restart after changing this. This is just a shot in the dark so still need the logs (unless this really does fix it). Edit: If your current "Connection Security" setting is STARTTLS, you might also try using SSL/TLS on port 993. Again, a tb restart recommended. "Normal password" should be fine.