My experience trying to use this program:
I was able to order a 7+ 32gb MB yesterday through the Apple Store App for same day pickup here in San Antonio. I paid full price through the app, then decided to try to switch to ATT Next at pickup to take advantage of this trade-in deal. I called apple support first to verify I could do this, and was told flat out that all I could do was cancel my order and lose the phone, then try to order another one. I declined their offer to cancel the order and went to the Apple Store for my appointment.
The Apple employee at the store also said that I was stuck paying full price, and my only other option was to cancel the order and try again whenever they got new inventory. She refused to cancel the order and then sell me the same phone through next, as some on here have been able to do. She did say that I had 13 days to return the iPhone if I found another one that I could buy through Next. I went ahead and bought the phone full price, just in case I was unable to find another one any time soon.
At 12:01 that night I found an Apple Store that had the same phone available (7+ 32gb MB) so I completed the sale through the Apple Store App for same day pickup, with ATT Next. Went to the store at my appt time, returned the original iPhone I paid full price for, and walked out with my new iPhone purchased through the Next Program. I then went to an ATT store, traded in my old 6s+, and was told the payments would start to show up on my 3rd bill.