He never got those services. On Dec. 6, the same day Dylan got into a private school program at United Cerebral Palsy, he was rushed to New York Methodist Hospital struggling to breathe. Two days later, his strength sapped, he went into cardiac arrest and died.

His case was one of thousands being fought each year as a growing number of families seeking private schooling at public expense have butted up against the city’s attempts to keep spending under control. This year, after complaints from parent groups reached state legislators, Mayor Bill de Blasio agreed to make changes that would streamline the process for some families, though he did not promise to challenge them less often.

For decades, parents of special-education students nationwide have been able to seek private schooling at no extra cost at public expense when public schools fail to meet their needs. Court rulings over the last decade have strengthened that right, and the city now spends more than $200 million a year on such education, up from just a few million dollars a year in the 1990s, when requests were far rarer.

Under the Bloomberg administration, the city beefed up its special-education legal team. The team evaluates requests and, increasingly, has denied them, arguing that city schools — where the cost of educating a special-needs student is generally $20,000 cheaper than in private school — are appropriate. Families can challenge the denial in a hearing. The Randalls’ hearing was one of 6,241 requested by families last year, up from 5,403 in 2008, according to State Education Department data. Families win a majority of those cases, according to city education officials. Either side can appeal a hearing decision to the state and, after that, in court.

In the past, members of the Bloomberg administration said they scrutinized requests to weed out families who were simply trying to get free private schooling, when public schools could meet their needs. Eric Nadelstern, a former deputy schools chancellor who is now a professor at Teachers College at Columbia University, said the efforts were rooted in a desire to pour money into special-education programs for all public school children, not to deny any one child’s chances for the best possible education.