I'm pretty sure there are documented, widely-recognized case of this happening. Can't remember the details, but I think it was an amateur radio operator that found a secret broadcasting station, and was thrown off the trail when said station decided to broadcast fake stuff about aliens. And another case where the government needed to test animals for radiation following a nuclear test, so secretly stole them from farms in a way that suggested UFO abduction?

I can't remember where I read/heard this, but sure it is on the internet somewhere.