This is no small matter, given that nonprofits accounted for 9.2 percent of all wages and salaries in the United States and 5.3 percent of gross domestic product in 2010, according to the National Center for Charitable Statistics. Given the many illustrious universities, hospitals, museums, orchestras, theaters and other arts organizations, nonprofits play an outsize role in the nation’s culture. But they have traditionally received little scrutiny until a scandal erupts or they’re on the brink of collapse.

Image “It’s easy to forget, but New York’s charities, collectively, are a big and important part of our state’s economy, and I consider it my responsibility to promote and protect the nonprofit sector,” Mr. Schneiderman said. Credit... Michael Appleton for The New York Times

State attorneys general oversee nonprofits both because they’re generally exempt from state taxes and because they represent the interests of donors who may lack the means to enforce the terms of their gifts or, once they’re dead, the capacity. (Since such institutions are also exempt from federal taxes, the Internal Revenue Service is charged with ensuring that organizations adhere to their tax-exempt purposes.)

Mr. Siegel said New York’s charities division “is really active, does a lot of investigations and really does go after people.” For most state attorneys general, he said, “their main focus in life isn’t charities and nonprofits.”

In large degree, this is a simple matter of capacity. The New York attorney general alone oversees about 80,000 nonprofits, ranging from world-renowned institutions like the Metropolitan Museum of Art and the Metropolitan Opera to little-known historic house museums. Given limited resources, “we have no choice but to go after the most egregious examples,” Mr. Sheehan said. He cited the example from last year of William Rapfogel, a close friend of the now-disgraced assembly leader Sheldon Silver, who was executive director of the Metropolitan Council on Jewish Poverty. He was sentenced to a prison term after pleading guilty to plundering the charity of more than $9 million.

But most cases of nonprofit mismanagement involve breaches of fiduciary duty, not criminal behavior, often rooted in a sincere if ultimately misguided effort to help. Board members are typically “people who are generous donors who support the mission of the institution,” Mr. Sheehan said. “There’s a culture of politeness and respect, and they support the chief executive. Most chief executives don’t want board members to ask tough questions. This is especially true when you have a charismatic leader.”

Cooper Union’s board took a series of financial steps that, with benefit of hindsight, seem misguided. The school borrowed $175 million for 30 years at a rate of 5.75 percent and then spent most of the proceeds on a lavish new building while continuing to run operating deficits. It also agreed to a prohibitively expensive prepayment penalty, making it financially impossible to extricate itself from the terms of the loan, according to people with knowledge of the agreement who spoke on the condition of anonymity. One of the issues in the current investigation is whether Cooper Union disclosed this potential penalty when it sought court approval for the loan.