I remember when a random selection of organizers were put on trial for "conspiracy" in response to the huge protest demonstrations at the Democratic National Convention in Chicago in 1968.
The government lost the case, correctly enough, because the defendants ~ a very loosely organized bunch of Christian pacifists, secular policial radicals, wild-eyed "Yippie" hedonists, and others ~ couldn't be proven to have collaborated on much of anything beyond a shared opposition to politics-as-usual in general and the war in Vietnam in particular. (It also turned out that some of the most visible and incendiary lawbreaking, notably a widely televised flag-buring in Grant Park, was actually the work of undercover agents provacateurs in the employ of government agencies.)
While there was in fact no conscious, well-planned "conspirancy" of the kind that many on the right were so anxious to discover and to prove, it was true enough that the disorganized, rag-tag group of protesters DID share a common purpose, and WERE "conspiring," but only in the most informal sense of the word. They shared a common sense of outrage and a common opinion of the status quo, even though the did NOT share many basic values and in fact held a wide spectrum of different opinions about how to bring about change.
The word "conspire" comes from the Latin, meaning literally "to breathe together." Those protest leaders in Chicago forty years ago did indeed "breathe together" insofar as they shared a common opinion about at least one critical issue and the need for masses of folks sharing that same opinion to express themselves in a mass protest. Any more delinite or more sinister collaboration on their parts could not be proved because it simply didn't exist.
I think the same dynamic, or somthing like it, is part of the unknown scenario behind most "conspiracy theories." For example, in the case of the JFK assassination, I find it impossible to believe that the CIA or FBI or any government agency ever actually made an official decision to kill the President, but I do find it entirely plausible that a few likeminded right-wing nuts might very well have found themselves, and found each other, in relatively powerful positions from which they would be able to act in concert. Nothing official, nothing on paper, and therefore nothing to go down in history, nothing for us to ever learn now that they're as dead and gone as the man they were so anxious to kill.