I'm going to add my 2-cents here as a nuke fresh out of Great Lakes. Sorry for the length in advance. I had a lot of time over the last 2 months to think about this topic :-)
Most recruits will be in a similar boat of a recruiter leading them to believe something "isn't important" or "doesn't matter". Many of those recruits will keep quiet through MEPS, Moment of Truth, and through special physicals and any other "opportunity" to fully disclose information. Some of those recruits will still make it through, and will still be able to get their clearance and continue their career in the Navy. Some of those recruits will have their past surface on its own and be shown the door. An RDC was shocked by how many he lost from his division for lying about Asthma - something that is waiverable even for nukes.
Personally, during the 7 days between taking the ASVAB and first learning about the Nuke program to signing the enlistment contract, I mixed the security clearance time window with the enlistment questionaire. This lead to me saying I had never smoked Marijuana at MEPS even though I had used it in high school. As I got further into my DEP process (10 months) and learned more about MEPS, enlistment, etc. I realized not only the mistake I had made, but also with the gloom and doom responses on the forum the possible severity of the consequences. That knowledge was extremely troubling. I was torn between speaking up, and losing an opportunity that I knew was the best I could hope for when I was at rock bottom, and losing my integrity and trustworthiness which are two qualities I consider most important in my life.
The last month - and especially the last week - before shipping out were filled with constant angst. I didn't want to speak up, get my recruiters in trouble (who did help me in every possible way throughout my time in DEP), and at best be pushed back further while a waiver went through, and at worst be kicked out before I even got in. But at the same time I didn't want to keep quiet about something so small and let it become a huge problem later on due to the breach of trust.
I kept quiet through MEPS, and got on a plane to Chicago. I kept quiet through Moment of Truth - I'd heard horror stories about this already and was certain that was the easiest way to go home. After the Moment of Truth the nukes were pulled aside and we had another "Moment of Truth" with the Nuclear Advisors (forget if thats exactly what they're called) which happened to be two Master Chiefs. After battling with myself for the previous 48+ hours I finally allowed my inner voice to take over. I made a committment at that time to disclose anything and everything that may have been ommited, glossed over, or I had been told was "unimportant." At that moment I decided that I would rather be kicked out of the Navy for telling the truth, than be kicked out for lying. From that point on was hell. I filled out a waiver request for the marijuana, told a few stories that had the Master Chief's eyebrows raised quite a few times but were verified to not require waivers or documentation, and was on my way. I was still wondering if that waiver request was going anywhere but Legal, and if I was going anywhere but home.
A week or so later, nukes have special physicals. That means after all the shots, pokes, and prods that you normally get in boot camp, you get to go back and get some more. During the brief the Corpsman gave an example that jogged my memory of being knocked unconscious 15 years earlier. I had honestly forgotten about the event as it was relatively minor, and happened when I was so young. I disclosed the discrepancy. I also disclosed the accident I had been in a few months before enlisting and was unsure if I had actually been diagnosed with something or not. Total honesty was what I had pledged to myself, so I was sticking to it. That lead to a less-than-pleasant experience where Article 83 was once again brought to my attention, and another few nights of little sleep during an already stressful period of boot camp.
A few more weeks down the line and I'm choosing my rate for the Nuclear program. At the end of the brief my name is called, "Your waiver has been approved." A very good day. A few more weeks down the line, I'm graduating and heading to SC, the records were located, documented, and it was verified that I was still fit for full duty. Another very good day.
Now I dont want everyone to think they can do the same thing. There was a change in command of the Nuclear Advisors office while I was there, and I believe there will be another one shortly (Master Chiefs are usually on their way to retirement...) so the same opportunity may not be available. However, I know people who came clean at MEPS, and still joined the Navy. I know people who came clean at Moment of Truth, and were still with us at graduation. I know I came clean, and I'm still in the Navy. So there is a chance to ensure your documentation is correct, and accurate without death and destruction. Is it a good chance? I don't know, I may have been lucky. But if I had it to do all over again I would have saved myself a ton of restless nights and gotten the proper documentation sooner, rather than later. It's not worth any additional stress when you go to boot camp, and it's not worth finding out that you love being in the Navy and want to continue your career, only to have a skeleton pop out and revoke your clearance 6, 10, or 15 years down the line.