If it helps anyone, the FiveThirtyEight article describes a scenario where people take a survey about immigration status and voting. Most legal citizens will correctly identify themselves, but some will accidentally check the wrong box and say they are an illegal immigrant. If you have a billion citizens and 10 illegal immigrants truly taking the survey, and people check the wrong box 1 in 1000 times, your "percentage of illegal immigrants who vote" statistic will be about the same as for citizens (because almost all reported illegals will be citizens). Collecting more data won't help.

It's a very good article, though in the context of deciding how many variables should be in a model of some complex phenomenon, this example is a little tougher to wrap your head around. It's not quite a predictive model, but there were some variables left out. A naive model I suppose is "this data is generated by infallible respondents", whereas a better model would incorporate that error rate. There isn't as much of a question about which pieces of information are relevant, though, like you might encounter when trying to predict future drug use from household income, race, age, number of books read as a child, number of pets, and so on.