I just thought I'd share what I'd figured out myself as I've had this problem for quite a while now,searched ALL over the net to find a solution and I ended up figuring it out myself
I have a Samsung Galaxy S3 (my second one!) I found that my text alert stopped working after a while,everything was turned on - sound etc.
On my first phone I thought it was a software/mp3 corruption issue so I reset the factory settings and noticed I was actually causing more problems,which is frustrating now I know how simple it is!
I have my own text alert rather than the Samsung ones (in mp3 format).
Anyway as I'm sure you're hopefully aware of how to use your own,if not just make sure the file is saved to "notifications" in "my files" when you go to "apps".
Sorry for all the typing just want to make sure I am clear
I found my S3 was working fine for the first week or so,I was getting my texts with the alert etc,then all of a sudden it would be silent but still appear on the screen,I would receive ALL my other notifications but the text alert.
Step 1 - Go to your messaging inbox and make sure you're on all the threads and press the bottom left hand button and select "settings" there you will see something called "storage settings" under "text message limit" you will see something like "200 messages per conversation" select this and change it to a higher number,mine is set to 5000 which I think is the highest.
Step 2 - Under the SAME section UNCHECK the "delete old messages" box,I have noticed if you don't this can cause the same problem because if the phone reaches it's set 200 it then tries to delete the older messages to make room for the new ones - hence the silence.
I did this last night and have had my text alert EVERY time now and all the way through I hope this helps everyone as I contacted Samsung and they advised me to do the dreaded factory reset which won't solve the problem!
Please let me know if it worked for you! x