Have you checked the text message notification settings yet? They're in a different location than the regular notification settings FOR SOME REASON.
To access those you can open the text message app then hit the menu button (bottom left corner) then you hit settings. If Notifications are off, you can turn them on. If they're on, turn them off for ten seconds and then back on.
If it's an Android device, are you using a 3rd-party App that's perhaps interfering with your Notifications?
Personally, I use an App called "Light Flow".
In addition to controlling the LED, it also handles the sounds & vibrations for all my Apps. Within the indiidual Apps themselves I switch off all their Notifications then use LightFlow to handle everything for me within that one App.