Yes. It happened to me -- not at USNA, but at USAFA, so there could be a difference.
I received "no" letters from all 5 service academies due to an issue with my medical review board. My USMMA recruiter (or Liaison Officer) was able to get to the root cause of the issue. This may not be the case now, but in 1995, I was taking allergy medicine and the review board failed me because of it. He instructed me to have my doctor submit a letter stating that I was no longer required to take the prescription medication and that I could use over the counter meds should my allergies flare up.
Once that letter was submitted, I received a medical waiver. At the time (not sure about now), the USCGA was unable to grant/use medical waivers. Within 24 hours of getting the waiver from the USMMA, the LO had secured my nomination from a neighboring district and it was a done deal. At least I thought for a few weeks I was going to USMMA. Once the review board had given me a waiver, I was back on the radar of the other service academies. That is when I received a call from my congressman's office indicating one of his nominees had backed out. It was 2 weeks before we were to report for in-processing and they asked USAFA who the next qualified person was on the list and it was me. I was then extended an appointment to USAFA and I elected to go there instead of the USMMA.
I did call the LO that made it all possible and even to this day I can not thank him enough for making it happen. I thanked him, and went to USAFA.
So, yes - there is a waiting list of qualified applicants. In cases where an appointee elects to go to a different school, they go to the next qualified applicant on the list.
Times may have changed, but I am certainly a lucky recipient of the qualified applicant pool.
Want a laugh? On in-processing day, you go through a medical screening. At that screening a doctor saw that I had a history of allergies --- and they offered me a prescription! Geez!