Heck, the FBI was involved in the one big conspiracy-coverup we know was true from the 20th century - and the UFO investigations were key to it.
When game-changing new airplane designs from stealth technology to the SR-71 were being invented, during the height of the cold war, it was all about secrecy - but what do you do about eye-witnesses to these odd-looking planes, including people like pilots that it's hard to write off as kooks? You investigate each reported sighting as a "UFO sighting", and then loudly deny that you're investigating UFO sightings.
The plan seems to have worked pretty well - eye witness reports and even some photographs of experimental aircraft were dismissed as fakes by the public - and as far as we know, by the Russians. The more interesting the reports - triangular aircraft with no tails, aircraft moving faster and higher than anything known - the more easily it was dismissed as "UFO nuts". Brilliant plan, really, and the only modern conspiracy I know of that actually kept a secret long enough to matter.