The usual conspiracy theory invokes what is called a "Grand conspiracy" in which other investigations, prior thinking on the same subject, and contradicting evidence are all dismissed as part of the conspiracy. This is laughably implausible because it involves this vast co-ordinated effort, often for no discernible purpose except to prevent the theorist from being right about something.

Actual secrets kept at some scale show the limits of what is possible, three of the best examples are from World War Two, a period that's near enough for us to have good records but distant enough that the secrets are long known to us.

Ultra is the most obvious. "The ultra secret" was the fact that Allied codebreakers could read some (though not all, at various points in the war) Enigma and other encrypted messages sent via radio by the sprawling Axis forces. Only a few dozen key people knew this secret, but of necessity hundreds, perhaps around 1000 more had some limited insight because their work was connected with the breaking of German messages (e.g. Hut workers at Bletchley engaged in translating broken messages or finding "cribs"). You will find people from the latter group who believed Ultra was protected well into the late 20th century, but in fact it ceased even to be an official secret after 29 years, and by the 1970s BBC documentary editors felt comfortable assuming that a general audience would know about Ultra in the television adaptation of Jones' autobiography "Most Secret War".

Second Manhattan. American development of a Nuclear Bomb could not of course be kept secret once the weapon had been used. But until then it was adequately guarded, there's no clear sign that Axis forces understood what the Americans were attempting or where. But lots of people tacitly knew about Manhattan, they just chose not to so anything until after the fact because "There's a secret military R&D base near where I live" is very obviously not something a patriotic person tells random strangers during a war...

Thirdly the Normandy invasion. Again after the fact this secret could not be kept. Within hours of making landfall the German high command were aware that this was not, as they had initially believed, a feint, and that all available resources must be diverted to defend against it. But that was far too late. Still in the weeks, and especially days, prior to the actual invasion, it will have been apparent not only to the soldiers involved but to nearby civilians that a massive naval landing was to be attempted. It just wasn't possible to hide from the people of Southern England that there were shitloads of infantry and boats making ready to depart in the next few days. Again the secret didn't get out, for a very short time, because it will have been obvious to all who knew that this wasn't something you should tell the enemy about.

Twenty Committee had successfully eliminated all German spies on the mainland (either turning them, imprisoning them, or as necessary killing them) so the only way for the secret to get out was for some idiot to tell the enemy themselves.

So those help you judge rough scale for a conspiracy. If a hypothetical conspiracy seems _harder_ to keep than, say, Manhattan, yet it seems to need more people to keep the secret than, say, Normandy, and for them to keep it longer than, say, Ultra, all because otherwise they might get sued, rather than because Actual Nazis are going to invade their country, well that's clearly crazy.