Every once in a while my device get "stuck" - I send an SMS or Email, and they are received only after 10 minutes or later. The device can be "stuck" even when the phone is on, and running other apps.
I tried installing Push Notification Fixer apps, but they do not help.
When I send the test and the phone is not "stuck", I get a notification after about 10 seconds.
However, I do receive notifications for Google Voice and WhatsApp when phone is in "stuck" phase, so it does seem app specific issue. Also I have noticed that these 2 apps have permission receive data from Internet , which the Gmail app does not. Seems to be related to what type of push notification is used. see documentation.
Also I noticed the Google Play Store is unable to install/uninstall apps in this "stuck" phase.