I think I might know what you are talking about. When I get a new text, I just get the regular alert. When I get a text in response to a recent text or a second text from that same person within a reasonable amount of time, the regular alert sounds with a little extra note or sound. I believe this is by design to imply that it isn't a new conversation but a response to a current ongoing one.
At least in my experience, the "extra" bit is not the same as the chosen notification so it isn't a true double notification.
Very hard to explain this without playing the sounds!