Re: New Jersey, New York
I'd bet a very small fraction of the US population can name the capitals of all 50 states. Probably the fraction that can name all the states in the first place is not large. Without trying it I'm not sure I wouldn't have a moment coming up with the lot.
There are quite a few states where the capital is not the best-known or most-prominent city. Michigan's capital isn't Detroit. Oregon's isn't Portland. Washington's isn't Seattle. Nebraska's isn't Omaha. Louisiana's isn't New Orleans. Missouri's isn't St Louis or Kansas City (though the latter would be a confusing choice, and both have the disadvantage of being located on the border).
The capital of Kansas is Topeka, which probably isn't the first city in Kansas that most people think of, though to be honest most people asked to think of a city in Kansas probably get stuck wondering whether Kansas City is in Kansas. (It is, if you're talking about Kansas City, Kansas. Kansas City, Missouri, isn't. And, yes, they're adjacent.)
Then there are edge cases like Texas - Austin is pretty well-known these days, but a few decades ago was more obscure outside the state. Or New Mexico; Santa Fe is actually the oldest continuously-inhabited city which is a US state capital, but most folks outside NM have probably only heard of Albuquerque and, for cultural reasons, places like Roswell and Los Alamos. (The latter not to be confused with the Alamo, which is in San Antonio, Texas. Remember the Alamo?) Or Vermont; it's hard to say whether Montpelier is as well-known (for smallish values of "well") as Burlington. Probably not.
And I don't know how many people on the street could, if pressed, name a single city in, say, Montana or Wyoming.