Have you tried restarting (turning it off and back on) the phone to see if this resolves it? Sometimes that is all you need.
Another thing to try is to make sure that the sounds for any apps you use are not interfering with the text message alerts. You may also want to check the phone's "Settings" and then "Sounds" and test the sound you have set for the messages to make sure it is actually still there and working.
Now if you are talking messages through an app (other than the phones regular text messages) you need to then check with that apps creators to see if there is an issue with it or a fix.