When something is returned as 'undeliverable', it is not Thunderbird stopping them. Thunderbird just downloads whatever is in the Inbox on server, so it has already arrived on server. It is the server sending the message because the server has rejected it. They may be 'undeliverable' because your email address on server has reached a full quota and cannot receive any more emails until some are deleted. So server will send a bounce back 'undeliverable' email. Check to see how many emails you have got stored on server. Are you close to the maximum allowed quota? If yes, you need to delete emails off server to create some space. Sometimes an email is rejected by the server because it fails the server spam controls. It might be being rejected because the sender was using port 25 which are commomly blocked by servers, so sender must use suitable protocols or perhaps sender was using an IP address that is blacklisted. That IP address could be their computers IP addess or the server smtp IP address. But only they can resolve that. Why it is undeliverable depends upon what additional information was offered in the 'undeliverable' bounced back email. Logon to your webmail account via a browser to see if there is anything in the server Spam controls for your account which has that email address as potential spam.