Has this happened to you?
New email messages may only appear to arrive in your Mail app when you first launch the app. No new mail seems to arrive until you actually quit and then relaunch Mail.
If this is a problem for you, Apple has released a new workaround that will get your mail coming in without having to quit the entire app.
Launch Mail and it will get your new email. If you want to check your mail later, and nothing has arrived, head up to the Mailbox menu and choose the “Take All Accounts Offline” option. Then, choose the “Get All New Mail” option from the same Mailbox menu.
If you don’t want to pull down a menu every time you want to check your mail, you can add a couple of buttons to the Mail app Toolbar.
First, choose Customize Toolbar from the View menu, and then drag both the “Take All Accounts Offline” button and the “Get Mail” button to your Toolbar. Then, you can just click each button in the same order as above and Mail will get your new emails without having to actually quit and relaunch the app itself.
It’s a silly workaround, but until Apple fixes the Mail app itself or those specific email providers change the way the email comes in, it’s the best we’ve got.