My son became a Depper in Sept 05. He was 17. In October (I think) he went down to MEPS, took the ASVAB the first day, and we met him down there the second. He passed the physical.
He, his father and I all met with the classifier for at least an hour. We did not even know of the existence of the Nuke program at the time, until they called in a nuke guy who came in and explained it to us. They said they thought for sure he would pass the test, but he needed to pick something else in case he didn't.
To be honest, our son at that time did not care what they gave him, he just wanted to be in the Navy. He picked something that he would be happy with, for 6 yrs.
A couple of weeks later he took the nuke test, passed, and signed new papers.
We were told later that he could back out up until the day he left for boot camp, though I don't know if that is true.
He has no record, traffic or otherwise, and I cannot think of one time during the DEP process where we as his parents have not been included when we wanted to be, or we have been lied to. His boot camp begins Aug 9.