I will get no notifications of new emails at all. It will only pull down emails when I go into the Gmail app on my N4, then it will pull them all down.
Can you tell if the emails are being pushed correctly, but you're just not being notified?
Or are the emails not being pushed (sync'd) until you open the app?
If it's just a lack of notification, then start the Gmail app and go to:
Menu - Label settings - Notifications - notify in status bar when email arrives (tick box).