Apple has posted a KnowledgeBase article with a workaround for persistent problems in Mail.app in Mavericks. Apple released a Mail update in November and fixed other problems with the release of OS X 10.9.1, but according to the KBase article, users are still having problems downloading new email from "some email providers."
The manual work around is essentially to take all of your accounts offline and then Get All New Mail. That will apparently force Mail to retrieve new messages. Apple also explains how to put the Take All Accounts Offline button on your Mail toolbar.
Apple's KBase article in full:
For some email providers, new email messages in Mail may only appear to arrive when Mail is first opened. No new email arrives until Mail is quit and reopened.
Quitting and reopening Mail forces it to re-establish a connection to the email server. You can use these steps to receive new mail messages without quitting Mail:
Choose Mailbox > Take All Accounts Offline.
Choose Mailbox > Get All New Mail.
As a shortcut, you can also add the Take All Accounts Offline and Get Mail buttons to your Mail toolbar.
Choose View > Customize Toolbar.
Drag the Take All Accounts Offline button to the Toolbar if it is not there already.
Drag the Get Mail button to the Toolbar if it is not there already.
To receive new mail messages, click the Take Offline button, then click the Check Mail button.
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