At Scarsdale High School north of New York City, one in five students is eligible for extra time or another accommodation such as a separate room for taking the SAT or ACT college entrance exam.

At Weston High School in Connecticut, it is one in four. At Newton North High School outside Boston, it’s one in three.

“Do I think that more than 30% of our students have a disability?” said Newton Superintendent David Fleishman. “No. We have a history of over-identification [as learning-challenged] that is certainly an issue in the district.”

Across the country, the number of public high-school students getting special allowances for test-taking, such as extra time, has surged in recent years, federal data show.

And students in affluent areas such as Scarsdale, Weston and Newton are more likely than students elsewhere to get the fastest-growing type of these special allowances, known as “504” designations, a Wall Street Journal analysis of data from 9,000 public schools found.