The boom began about five years ago, said Kathy Pelzer, a longtime high school counselor in an affluent part of Southern California. More students than ever were securing disability diagnoses, many seeking additional time on class work and tests.

Such psychological assessments can cost thousands of dollars and are often not covered by insurance. For some families, the ultimate goal was extra time — for classroom quizzes, essays, state achievement tests, AP exams and ultimately the SAT and ACT.

“You’ll get what you’re looking for if you pay the $10,000,” Pelzer said, citing the highest-priced evaluations.

From Weston, Connecticut, to Mercer Island, Washington, word has spread on parenting message boards and in the stands at home games: A federal disability designation known as a 504 plan can help struggling students improve their grades and test scores. But the plans are not doled out equitably.

In some communities, more than 1 in 10 students have one — up to seven times the rate nationwide, according to a New York Times analysis of federal data.

In Weston, where the median household income is $220,000, the rate is 18 percent, eight times that of Danbury, Connecticut, a city 30 minutes north. In Mercer Island, outside Seattle, where the median household income is $137,000, the number is 14 percent. That is about six times the rate of nearby Federal Way, Washington, where the median income is $65,000.

Students in every ZIP code are dealing with anxiety, stress and depression as academic competition grows ever more cutthroat. But the sharp disparity in accommodations raises the question of whether families in moneyed communities are taking advantage of the system or whether they simply have the means to address a problem that less affluent families cannot.

While experts say that known cases of outright fraud are rare, there is little doubt that the process is vulnerable to abuse. Some of the learning differences exist in diagnostic gray areas that can make it difficult to determine whether a teenager is struggling because of parental and school pressure or because of a psychological impairment.

Many Americans got their first look at disability accommodations in the wake of the college admissions scandal, in which affluent parents were accused of hiring a consultant, William Singer, to cheat their children’s way into prestigious universities through a variety of schemes.

An Unequal Diagnosis

The 504 plans, which get their name from Section 504 of the Rehabilitation Act of 1973, are intended to help people who have a physical or mental impairment that “substantially limits” learning or other activities. They offer students such accommodations as a seat at the front of the classroom or a private room for exams, free of distractions.

One of the most common accommodations is extra time on classroom tests, which the two main college admissions testing companies, the College Board and ACT, look for when determining whether to grant students additional time for their exams. Many students struggle to complete standardized tests in the allotted minutes, and research has found that having more time can raise scores for students who have a decent grasp of the test material.

The testing agencies also look for detailed diagnostic evaluations conducted within the last several years.

In an analysis of Department of Education data, the Times looked at students with 504 designations at more than 11,000 high schools across the country. It did not include students who are served under the Individuals with Disabilities Education Act, a further-reaching program that is generally meant for students more severely affected by disabilities.

The Times found a glaring wealth gap in 504 designations. At high schools in the richest school districts — the top 1 percent as measured by census income data — 5.8 percent of students held a 504 plan, more than double the national average of 2.7 percent. Some wealthy districts had 504 rates of up to 18 percent.

There were also racial disparities, the Times found. A larger percentage of white students held a 504 plan than students of any other race.

Federal disability data does not include private schools. But in some areas, like Manhattan and the West Side of Los Angeles, private school students are even more likely than affluent public school students to use disability diagnoses to qualify for extended testing time, according to research and interviews with school workers.

Looking for an Admissions Edge

Pelzer, the California high school counselor, said the frenzy over college applications was a major cause of the growth in disability diagnoses. “It’s the competitiveness and wanting to get the edge,” she said. Private admissions consultants, she added, recommend that parents seek out diagnoses, and word gets out.

Online message boards are another wellspring for tips. On one serving the capital region, DC Urban Moms and Dads, a parent inquired in January about how to get accommodations for a 10th grader with a 3.6 GPA who received an ADHD diagnosis in elementary school but was denied a 504 plan. “We started thinking ahead to junior year and beyond when the coursework will undoubtedly get harder,” the parent wrote.

The College Board said that in recent years 4 percent of the roughly 2 million students who take the SAT each year used accommodations, up from 2 percent in 2002. According to ACT, 5 percent of the 2 million test-takers in the high school class of 2017 used extended time on its tests; the company said it had not tracked how the percentage had changed since the early 2000s.

But not having an up-to-date evaluation can hurt a student’s chances.

Melanie McDaniel, 21, was in the fourth grade in San Antonio when her parents learned she had ADHD. By her junior year of high school, she was struggling so much during tests that at the sound of a pencil dropping, she said, “my head would be spinning.” Her school gave her a 504 plan that offered her extra time on exams.

But when McDaniel applied for more time to take the SAT, she was rejected. Her parents never got a concrete reason, but her ADHD evaluation was 7 years old and did not include a full battery of assessments.

Reapplying with a new evaluation would have been a stretch for her middle-class parents. “My mom said we couldn’t afford it because it cost thousands of dollars,” McDaniel said.

Inside the Psychologist’s Office

“Get ACT Extra Time,” reads one blunt web advertisement from the Cognitive Assessment Group. “35+ Years of Accurate Testing.”

The New York-based practice behind the ad is run by Wilfred van Gorp. In his current practice, van Gorp assesses 20 to 24 patients a month, he said, two or three of whom are high school students seeking a diagnosis before they sit for standardized tests. The appointments cost about $6,000 and are often not covered by insurance, although he said he provided financial assistance to some patients.

While abnormal scores on these tests may lead to an ADHD diagnosis, van Gorp said he also had the discretion to give a diagnosis to a patient with normal scores if he observed signs of inattention, like constant fidgeting.

About 70 percent of the patients he sees leave with a diagnosis, he said, and testing agencies usually approve accommodations for them. He acknowledged that some patients arrive with information about the assessment process, gleaned from internet research. But he said he tried to screen out cheating through tests of effort and motivation.

The ability to rehearse for assessments “is a concern both of mine and the field in general,” he said.

Lorrie Ann Ness, a psychologist in Silver Spring, Md., outside Washington, sees teenagers who “are working 300 percent harder than their peers to keep their heads above water,” she said. “Their tears are real. Their pain is real.”

Like several psychologists interviewed across the country in the wake of the college scandal, Ness said some practitioners conduct rushed assessments that could result in a misdiagnosis. “There is some really bad, sketchy stuff out there, and schools are right to be skeptical,” she said.

In the college admissions fraud case, the government is not investigating any psychologists to whom Singer referred clients, according to a person with knowledge of the investigation, who requested anonymity because of the sensitivity of the case. The American Psychological Association said it was “not aware of any widespread, systemic” abuse of the systems for diagnosing learning differences or recommending testing accommodations.

Rachel Fish, an assistant professor of special education at New York University, said it was difficult to determine whether wealthy and white students were being overdiagnosed with conditions such as ADHD and anxiety, or whether poor students and children of color were being underdiagnosed.

“It’s different levels of advantage in being able to acquire services,” she said.