The problem is that there's no reasonable way to grade the quoted rejects as false. If you aren't a lawyer (Edit: but maybe if you are*), there's really nothing about labeling John Roberts as Chief Justice of the Supreme Court that is more useful than labeling him as "the justice in charge of the Supreme Court." The error is roughly on par with asking "what does 2+3 =" and accepting "V" but rejecting "IIIII"
In short, I have dramatically adjusted downward my belief the reliability of public-ignorance surveys.
On reflection, I think some of the answers could be considered wrong in a technical sense not relevant to the question being asked. For example, "in charge" implies a bit more power over Supreme Court decisions than Roberts actually possesses.
In the old version, I stated that the difference wouldn't matter even to a lawyer.
I haven't. I expected they were making mistakes like this one, and haven't seen anything indicating they generally make mistakes in this direction rather than the other.
It makes sense to adjust downward your belief that they are reliable, if you thought they were very reliable before. But this shouldn't be enough to indicate they're reliably getting it wrong in a particular direction.
How many times have you heard a claim from a somewhat reputable source like "only 28 percent of Americans are able to name one of the constitutional freedoms, yet 52 percent are able to name at least two Simpsons family members"?
Mark Liberman over at Language Log wrote up a post showing how even when such claims are based on actual studies, the methodology is biased to exaggerate ignorance:
If, every time you heard a claim of the form "Only X% of Americans know Y" you thought "there's something strange about that", then you get 1 rationality point. If you thought "I don't believe that", then you get 2 rationality points.