Apple Support Offers Workaround For Issues

Apple Support:

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.

What an embarrassment. Just fix the thing. iOS 7 Mail is buggy too, occasionally refusing to mark messages I am currently reading as read.