To supplement referral to the "foundation to usna thread": If you rec'd a 3Q letter, you're probably not a candidate for a service academy prep school.
Another answer to your question (Or can candidates that are fully qualified be given a slot as well?) is "sort of." There was a poster on this site - a female, I think - who felt as if she rec'd NAPS because she wasn't physically fit, though she was a competitive candidate academically. So apparently she wasn't fully qualified in terms of PT.
You can read elsewhere that the SA's prep schools (not to be confused w/ the private and foundation schools) are primarily intended for (a) prior enlisted (b) athletic recruits (c) under-represented minorities, and (d) applicants who have something the Academy likes, but aren't quite academically competitive in terms of test scores OR GPA .
This question often invites input from the racially-rabid posters, but I'd like to speak for group (d), since that's where my white male Midshipman fell. The academic question of "fully qualified" varies each year, since the norm moves w/ the applicant pool. As I've understood it, applicants who are "rejected" from USNA candidacy are put in a pool to be considered for prep school. Then it becomes a question of what the candidate brings to the party. Only a couple hundred are offered that extra year on the Navy's dime. My son (and I) believe he was offered NAPS because, while he had a great app in terms of GPA from a competitive school, leadership, and EC's, his test scores weren't quite where USNA likes them. He also had a great BGO interview (w/ an awesome BGO), and wrote a fine essay.
If you search this thread, you'll find a parents who are outraged that their 3Q'd kids were "more qualified" than kids who were offered SA prep schools. The Academies seem to know what they're doing. My former-NAPSter will graduate in May in the top 15% of his class, already in grad school via the VGEP program, lots of leadership @ USNA, etc. Plenty of h.s. valedictorians @ USNA and perfect SAT Mid's weren't as fortunate.
I hope this helps answer your question.