Are these POP or IMAP accounts? When you say some accounts stop downloading, have you checked the account via webmail and found new mail that isn't arriving through TB?

They are POP accounts. Yes, the webmail is downloading all the emails.

Hi I am not sure this will help but there seem to be several people reporting similar issues. This one is recent: https://support.mozilla.org/en-US/questions/1273921

The bug is when Local Folders is the default account and prevents downloading at startup, which is the opposite of this problem. What is the time between polling of the POP server? If it's too short, the server may reject downloads after the one that succeeds at startup.