I tried to replicate the double ring tone again on my 4S with the FaceBook app as per how you did it, but unfortunately I'm unable to get the double alert again.
I doubt it has anything to do with the tri-tone because you can set the text tone to whatever you want and I was having the problem with a different tone and not the tri-tone.
My theory is that it has something to do with the way how iMessage is delivering the messages via the internet -lag perhaps? I did notice that when I was having the issue, my text messages were coming in late and in hordes at a time with the annoying overlapping.
For those who had the iMessage feature turned off from the start. When the iPhone was first setup on iOS 5, it tries to enable the iMessage feature by default. Try enabling the iMessage, by this I mean doing it right with a complete activation login with email -not just turning it on and off, reboot, then turn the feature off, reboot again, do a text test and see if you still get the overlapping tone.
Let me know if this fix works for you. I want to duplicate the problem again, but it will not come back =/