One company I worked for had a let down into a private HLS. It involved flying over an NDB on a local airfield. When we used it we always called on the airfield frequency, for obvious reasons, and always gave way to traffic in the pattern or joining.
Two colleagues of mine (two pilot ops, both very experienced, long term ex-military pilots) once watched slightly bemused as a light aircraft flew under them; no radio call being heard from it. They assumed at first that it was carrying out a non-radio overhead join, so they gave way to it. But it maintained altitude then turned ninety degrees away from the field and departed again.
After landing they received an angry phone call from an instructor. He was livid that his student had been inconvenienced during his circuit practice. My colleagues couldn't understand how this aircraft had been in the circuit from the way it had been flown. As the conversation continued it became apparent that the aircraft was actually practicing so-called "circuits" at another airfield six miles away!
My colleague (he was the chief pilot of our company) told the instructor to file an AIRPROX so the radar tapes could be examined. The instructor went very quiet and that was the end of the conversation, no AIRPROX was ever filed.