The same thing has been happening to me too since I downloaded iOS 9.3, and the problem wasn't fixed in 9.3.1. Mine does what y'all are talking about, but then it even goes a step further and messes up the notifications when they're read if I double text. For example
4:12 "blah blah blah"
- read 4:16
4:14 "blah"
- delivered
Even though they were both sent before being read at 4:16, only the first message says that it was read.
I Contacted Apple about this the other day and did the following troubleshoots
- turn off iMessage, hard reset (hold and home) - no avail
- reset network settings - No avail
- delete whole message thread with the contact it was happening with - no avail
the apple supervisor suggested that it was the recipients phone. He did all the same troubleshooting steps I did and nothing worked. I'm waiting for the apple supervisor to return my call with the next step, but I'm glad I'm not alone! Being on the phone with them for an hour started to make me feel like a crazy person and like I was making it up!
I'll send updates as my personal saga continues in hopes that it'll help all of you suffering with this problem