I have an iPhone 6s running iOS 9.3.1. I have notifications for messages configured to alert with a sound. Other apps are set to notify with sound as well. I do have Do Not Disturb enabled, but it is on a schedule (nights only). The mute switch on the side is turned off. Phone calls and other things make noise properly.
When I receive a text and the phone is sleeping, I get no vibration, no banner popup, no sound. If I then press the power switch, I see the message notification on the screen, so I know the phone did receive the message.
Same issue with the Apple watch I have paired to the phone. I see that the text arrived, but no alert.