Problem with iMessage "Delivered" status
Basically, your message has been sent, from your end, to another iPhone or Apple apparatus (it appears in Blue and not Green) but has not been delivered to your recipient. 

"Delivered" may not appear for several reasons. The most likely, and also the only I've confirmed, is that the recipient cannot receive the data. This happens when a phone is off or out of service range, obviously. It also happens frequently if they are talking on the phone. When someone is talking voice, and out of a WiFi area, they may not be able to receive text messages, because they may drop or lose some of the data, so it is conveniently held onto for them until they're freed up (i.e. off the phone). It can also happen when WiFi access is spotty, when other apps are running in the background, etc. 

The reason you may be experiencing this but still receiving responses is that once a response is sent (in any exchange) that "Delivered" or "Read" message vanishes. Poof. So you're probably just not seeing it change to Delivered to Read before you see a response. 

Another possibility that makes sense and could explain both, however unconfirmed by yours truly, is that the "Do Not Disturb" feature is toggled on the recipients phone. As far as I was aware, this simply prevents them from receiving notifications of the messages, but I haven't test it out.  

It's one of the above anyway for sure.  Hope it helps.