Q: No sound on iPhone when message arrives

Varify your settings  If you hear no sound or distorted sound from your iPhone, iPad, or iPod touch speaker - Apple Support

especially your control center  Do not disturb is not active:

Do Not Disturb: Silence calls, alerts, and notifications that you receive while your device is locked.

I'm having this same problem after updating to iOS 9.3.2, I don't get any vibrations or notification sounds from my iPhone 6+ when the screen is locked.

If I have my screen unlocked then I get vibrations and notification sounds as expected.  "Do Not Disturb" is not enabled, and even if it were I have the option set to keep silent when unlocked so this is not a setting problem but a bug in 9.3.2.

I've powercycled the phone, no change.  Notifications to my Apple Watch are still working fine.

My issue is related to my Apple Watch, if I turn my watch off then I get notifications on my phone as expected. Turn my watch on and notification sounds/vibrations stop happening on my phone. Both my watch and phone are running the latest versions.

I've confirmed that this seems to be the correct/expected behavior and there is no way to disable it.

Remove the watch from my wrist so that it locks but is still connected to my phone and alerts happen on my phone as expected. Put the watch back on my wrist and the notification alerts only happen on my watch

Sigh, thanks Apple.