In the wake of the 2016 election, Trump claimed to have had higher turnout at his inauguration than Barack Obama did. Subsequent polls and surveys presented people with pictures of Obama and Trump’s inauguration crowds and asked which was bigger. Republicans consistently identified the visibly smaller (Trump) crowd as being larger than the other. A narrative quickly emerged that Trump supporters literally couldn’t identify the correct answer; they were so brainwashed that they actually believed that the obviously smaller crowd was, in fact, larger.
Of course, a far more obvious and empirically plausible explanation is that respondents knew perfectly well what the correct answer was. However, they also had a sense of how that answer would be used in the media (“Even Trump’s supporters don’t believe his nonsense!”), so they simply declined to give pollsters the response they seemed to be looking for.
As a matter of fact, respondents regularly troll researchers in polling and surveys – especially when they are asked whether or not they subscribe to absurd or fringe beliefs, such as birtherism (a conspiracy that held that Barack Obama was born outside of the US and was legally ineligible to serve as president of the United States).
This is an absolutely vital point.