(Android apps are not as friendly to a "WiFi Only" mode.)
Let me expand upon this statement. You commented that you turn your Cell Data off. That could be the problem. You see, on Android, the signal from the cell data is what tells the phone to wake the WiFi and get the notification. As I understand it (Like I said, I refuse to use any Apple junk), the iOS devices power down the WiFi, and occasionally power it back up to check for notifications.
I have a WiFi only Android tablet. It can be sitting here, on my desk, screen off, and I won't get my notifications (email or anything) on it until I wake the screen up and it reconnects to WiFi.
My Atrix, laying right beside it, with the screen off, neither one of them touched in the same amount of time, gets the alerts. Why? It got the original "Ping" that there was an alert through the 4G, woke the WiFi, and pulled the full body of the notification over WiFi.
Try NOT turning your carrier data off and see if that does not resolve the problem. FYI - I never turn my data off, I have 2 work accounts, a gmail account, a yahoo account, a RSS reader, Facebook, UberSocial... and several other apps that get notifications. I never turn my carrier data off, I never miss a notification (except for Facebook, but that app has never given notifications properly). And, according to 3G watchdog, I used less than 200MB of data a month. That includes when I actually read/respond to emails, read RSS and access websites on the go.
I don't know why I pay for the 5Gb unlimited data anymore. But, I guess it's because I'm grandfathered in and don't want to give it up!!