There are two mechanisms by which conspiracies were described to be impossible in the past. Hanlon's razor, and the impossibility of shared secrets.
Hanlon's razor has been going obsolete since the industrial revolution. The probability of you interacting with someone is correlated with their influence. Influence is correlated with net-worth, which is correlated with competence, and anti-correlated with benevolence. Furthermore, competence is not a conserved quantity. You can now hire people to extend your competence by paying them to write algorithms, copying a piece of their competence and giving it to you.
It turns out that it's neither as hard nor as necessary to keep shared secrets as it was originally thought. Encryption, Need to know protocols, and the extension of our competence and influence with computers and networking has reduced the number of people who are required to know the full secret to very nearly one, despite wide distribution of the contents.
And while inference is still possible on the secret fragments, it is very difficult to communicate that inference to other people in a way which they find believable. This is partly because of the non-uniqueness of inferences, which should be irrelevant, since this is how all science works. The other part is that people are largely not equipped to understand those inferences. That sounds like a polite way of calling them stupid, but it's really just that even, extremely detailed schemes with nigh unobservable effects can be distributed on a global scale, with extraordinary benefit to the perpetrator. It's computationally infeasible to have any proportion of the population be aware of society to this level of detail.
As a glimmer of hope, something that we are capable of, however, is determining the trustworthiness of people. We don't seem to elect leaders based on their trustworthiness anymore, but it would not require a fundamental shift in human competency or nature to move to that system. It would require a more complete understanding of what trust means, though.