Text messages don't beep at me:
- Yes, the volume is up (I can hear when people call me)
- No, it's not quiet hours
- Settings below. Is there anywhere else I should look?
If the sound plays while the phone is unlocked, but not when locked, notifications might be disabled on the lock screen. To change this:
Note that this will do exactly what the text suggests - in addition to the sound being played, the notification itself will also be shown. If you wish to prevent this (for obvious privacy reasons), you'll need to change the notification settings for each app to enable the "Keep notifications private on the lock screen" option. For more information about that option, see e.g. this article.
Allow Messaging app to run in background
If you have both Messaging and Messaging + installed, try uninstalling one. It worked for me.