This happened to me recently too - but I'm not sure if it's the same thing that's occurring with you. First off, I think it started after an update and happened to me and some friends.
For me, the problem is if I leave the text messaging app open in the foreground, with the specific user message thread open. If the message thread is open (even if the phone is in sleep), then it will not notify you when there is a new text message from that person. If you back out into the main text messaging screen (where it shows all of your texts), then it does notify me. So my solution is to not keep the messaging thread open for a specific person. Hope this helps.