But over time, they and their neighbors at 501 W. 143rd St. learned that this experiment in home ownership for the poor sometimes has a downside, and now they are working to save their building—and their investment.

While many of these limited-income co-ops have been successful, nearly a third are struggling to pay tax bills, which is a sign of poor overall financial health, according to data provided by the city’s Independent Budget Office.

The city has about 1,000 limited-income co-ops, which over the past five years have accounted for nearly half of all delinquent tax payments from the city’s more than 4,800 co-ops.

Many housing experts say that in problem buildings, management is often lax, maintenance fees are kept too low to pay bills and care for buildings, and many are run by fiefs of shareholders around whom swirl allegations of favoritism and corruption.

“They view themselves as tenants rather than as apartment owners,” said Steven R. Wagner, a co-op and condo lawyer, who often warns buyers about the risks of buying what are known formally as Housing Development Fund Corp., or HDFC, co-ops. “They are not used to the responsibility of managing a building or running what can be a multi-million-dollar business.”

Last month, the Kiels and their neighbors received a city “notice of possible foreclosure” citing $3.2 million in unpaid water bills, taxes, interest and penalties, totaling about $85,000 per apartment.

The co-op had accumulated hundreds of violations and had stopped holding annual meetings, distributing annual financial statements or paying bills from the city.

“The accountant was the first to go,” said Yvette Hanon, a managing agent who has run the building for two decades. She said what was happening to the building was “very depressing” but that there just wasn’t enough maintenance revenue coming in to do much more than pay the heating bill.

Starting in 1981, the city created more than 1,000 low-income co-ops from a backlog of abandoned buildings seized during the city’s 1970s financial crisis.

Christopher Allred, an assistant city housing commissioner for asset management, said the co-ops were “largely a success story.”

“Most of the co-ops are well-functioning, affordable housing resource for their shareholders, and they have stabilized thousands of lives,” he said.

Over the years, he said, the city has increased training for residents of new co-ops and spelled out the obligation of co-ops in formal agreements with the city. Ms. Cheng had asked the city to investigate 501 W. 143rd, but Mr. Allred said it doesn’t have the authority to intervene.

“We take a tough-love approach,” he said. “They are the owners, and as the owners they are responsible for making decisions on their property.”

At the request of The Wall Street Journal, the city’s Independent Budget Office compiled five years of city property tax bills for a list of limited-income co-ops. The analysis found that by June 2014, 32.8% of the co-ops owed at least one year’s taxes, 26.9% owed two years’, and one in eight owed five years’. The tallies exclude buildings that were billed less than $100 a year in taxes.

Of all taxes, interest payments and other fees billed during the fiscal year ending in June 2014, 37% were unpaid by the end of the year, far more than other types of building. In 2014 the city’s finance department put the property-tax delinquency rate on co-ops at 0.5%, and for condominiums at 2.5%.

The building at 501 W. 143rd has 38 apartments and faces a triangular park and the wide intersection where Hamilton Place meets Amsterdam Avenue. It was known as Hamilton Court when built in the early 20th century, and boasted mahogany doors, parquet floors and dining rooms with panels of Flemish oak, according to a 1908 publication, “Apartment Houses of the Metropolis.”

Unhappy shareholders held what they called a “rent strike” and filed suit in 1998 to get control of the board, with the help of legal-aid lawyers. But they lost after a judge ruled that because they were in arrears on their maintenance, they weren’t eligible to vote. There was another failed attempt in 2007.

Robert Cochran, a disabled Vietnam War veteran, said he had been battling the board for years. Hot-water pipes in Mr. Cochran’s bathroom wall have been broken for months. One floor down, Clement Nichol, a retired doorman, has similar problems: His three children—Justin, 14; Marcus, 10; and Emily, 13—wash up in a blue plastic bucket because the shower doesn’t work.

The Kiels filed suit against the board and the managing agent in 2013, seeking appointment of a receiver to take over operation of the building. Ms. Hanon declined to discuss the case, but court records show that in February she was found in contempt for failing to turn over co-op documents.

If the city forecloses, the building would be turned over to a not-for-profit, then renovated and turned into a rental building, city officials. Owners would lose their equity but get to stay in their apartments.

City Council member Mark Levine, who characterized the West 143rd Street building as the most troubled HDFC in his district, said he was working with building residents to “help them form a working board.”

Michelle Smalls, the current board president, declined to comment, except to say that she is worried about the future of the building. “I live here too,” she said.

Ms. Hanon said there is plenty of blame to go around. She said the board had raised maintenance only twice in 25 years, while owners acted like tenants and called the city, for repairs racking up violations on the apartments they own. There are now more than 300 housing violations on the building.