From Weston, Conn., to Mercer Island, Wash., 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 across the United States.

In the country’s richest enclaves, where students already have greater access to private tutors and admissions coaches, the share of high school students with the designation is double the national average. In some communities, more than one 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, Conn., 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, Wash., 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, and that most disability diagnoses are obtained legitimately, 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. And private mental health practitioners operate with limited oversight, either from school systems or from within their own professions.