iPhone Notifications Not Working

I've got a strange problem with my iPhone 4, which I didn't notice until after I had set up my iPad. It seems that apps on my iPhone aren't sending notifications to the lock screen/notification center in most cases (I noticed this when notifications started pouring through on the iPad but I got nothing--no lock screen, no notification center, no badge, no nothing-- on the iPhone. I've checked the notification settings for the apps and they mirror the iPad: Notification Center, Badge App Icon, Sounds, and View in Lock Screen all ON.

Sometimes badges work, sometimes they don't. It seems that badges only consistently appear on the app *after* I've opened it.

I've tried doing a restore on the iPhone, but it seems to have not made any difference. Has anyone experienced anything similar?

 

Lots of views but no feedback... nothing? I've verified that in some apps push notification just isn't working at all. I have set alarms in some apps that are supposed to go off at a certain time but nothing happens.

I'm about to try Resetting my phone and setting it up as a new phone. That is, not restoring from iCloud backup. I'll lose all of my app data but without any additional feedback I just don't know what else to do. I've restored from the iCloud backup and it's still doing the same thing.