There are a lot of software "glitches" - what a senior person/manager at the genius bar told me.
I have this problem also. Have talked to reps twice today - since it worked when they were on the phone, even though I showed a picture from 30 min later when it didn't work and gave them detailed information, they said well is it not happening now so you are ok. I don't know what is happening to support but not like it used to be.
Make sure that under settings, Messages, you have imessages turned off. Use regular SMS messages instead. In my case it was trying to send out an imessage to a phone that didn't have imessage turned on so delays would happen from 30 min up to 10 hours. And doing this despite imessages being turned of on the phone. If you are not using facetime, you might try turning that off also.