Is it possible for someone from NAPS to not get a nomination, despite meeting all requirements?
There are a number of factors that go into the NOM, however, it would be possible under some circumstances for the candidate to not receive one from their Congressional or Senatorial pool -- but it isnt common and if this should happen, there are other ways to get the candidate a NOM.
USNA clearly has a vested interest and (demonstrated) financial commitment into the candidate. Assuming there was not a discipline, academic, or honor issue with the candidate thus making the SA no longer interested in the candidate, the SA will find a NOM for the candidate if the candidate is not selected from their Congressional or Senatorial pools. This could mean they grab an unused slot from another district or state, or it could come from Superintendent, or Vice Presidential.
There is an element of political capital value to the MOC for having as many constituents benefit from their assistance as possible. In terms of NOMs, MOC boards balance between rewarding local candidates and pushing forward the most competitive candidates who have a likely chance of appointment. In competitive districts, it is common for the MOC to physically exceed the 5 student (in their name) limit at any one SA. This is done by filling seats from other MOC districts who are either not qualified or do not submit (weird but it happens). If the MOC sees a sponsored prep school student within the pool, in my experience, this is viewed as the student WILL RECEIVE AN APPOINTMENT if nominated.
In my experience, the board wants as many students to receive an appointment as possible from their district and will nominate those who are viewed most likely to receive an appointment. Once those slots (of the 10 per cycle vacancy - aka slate) are filled, the remaining names are added based on a variety of criteria. The top criteria, in my experience, was appoint-ability based on the input of the MOC board members (typically alumni of each SA).
Historically, Blue Chip athletic recruits are viewed the same way - they will get an appointment. However, I have seen cases where a MOC elects not to nominate the recruit because of a strategy that suggests the SA will find a different path for this candidate because they REALLY want the candidate in the next class and will use a Superintendent NOM or other nominating source. It sounds weird, but if you have a pool of VERY competitive candidates, and this recruit simply does not match up (outside of their demonstrated athletic abilities) to the others in the pool - the MOC may opt to nominate their top candidates and not the recruit.
It is a weird game of numbers behind the scenes and to your original question, it is possible for a sponsored prep school student to miss out on a MOC NOM, but not something that happens often.