Apple Support Offers Workaround For Mail.app Issues

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.

Resolution

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.