I have discovered a settings change related to this. Since iOS 9 I noticed I was not getting banner notifications but only sometimes. On those occasions even changing the notification type from banner to alert made no difference.
I finally figured out it was when my was phone was in do not disturb mode. There's a dnd setting that says are notifications silent only when the phone is unlocked and in dnd mode, or are they silent always in dnd mode.
My setting in iOS 8 was always. Thus after midnight, when my scheduled dnd mode starts, if I'm still up and using the phone, incoming messages won't beep and wake my wife. But I would see the banner notification. This worked great for me.
In iOS 9 this same setting works differently.
If for dnd you select silent always you no longer get any banner or alert notifications. The app icon badge will show (if that's selected for the app in question).
In the case of messages this is a terrible change. With iOS 9 if I'm in another app and in dnd mode with silence always, I have no idea if a message has come in. If I change the silence setting to only when phone is locked then I do get a banner notification but I also get the sound of an incoming message.
Thus I'm forced to manually switch the phone to silent although it still vibrates and I have to remember to switch it out of silent when locking it for the night.