If you find that mail will suddenly not download, when it was downloading before, it could be due to a few reasons:
1) You have Postbox set to check for new messages too often, and the server isn't responding. Try setting the time interval to check for new messages to 15 minutes:
Change How Often Postbox Checks For New Mail
2) You have too many devices (computers and phones) checking for mail on the server at the same time, and the server isn't responding. Try closing down some email apps.
3) You could have too many connections to the server. Try lowering the number by following the instructions in this help article:
Too Many Connections to the IMAP Server
4) Finally, try deleting your caches folder:
How to Delete the Postbox Caches Directory
Please remember to restart Postbox after performing any changes.