A week later, Ms. Gilman’s attorney was told that the board wanted Ms. Gilman to put two years of maintenance payments, around $46,000, in an escrow account indefinitely. Her lawyer told the seller’s attorney that Ms. Gilman would respond upon her return.

It is not uncommon for a board to make such a demand, according Marc J. Luxemburg, a partner at the law firm Gallet Dreyer & Berkey and the president of the Council of New York Cooperatives and Condominiums. He put it this way: “It’s an alternative to being rejected.” If a board is on the fence about you, it could make such an unpalatable demand to encourage you to walk away.

“That’s a lot of money,” Ms. Gilman said, speculating that “they just wanted to keep it forever.” Regardless, she never had a chance to turn down the offer. The board rejected her application before she returned home, saying that she did not respond in a timely manner. “I just want to strangle them,” she said.

Then the seller would not return the deposit, claiming damages. “What damages?” she said. “I was in their apartment two or three times.” According to Mr. Luxemburg, sellers are increasingly holding on to deposits when deals fall apart, forcing the buyer to sue to get the money back or settle. “It’s a more aggressive approach,” he said. “But it’s becoming more common.”

To avoid litigation but still get the bulk of her deposit back, Ms. Gilman agreed to let the seller withhold two months’ maintenance, about $4,000, and she received the balance of her deposit. “These people have treated me abysmally,” she said.

The co-op board and its property manager, Brown Harris Stevens, declined to comment.

Boards generally reject buyers for financial reasons. But these decisions often do not come down to bank statements alone. Co-ops are vertical neighborhoods with unique personalities and cultures. A board that is looking for a certain type of person can glean information from financial documents about a candidate’s political leanings, health problems and marital strife.

“There are cases in which it’s a total mystery,” Mr. Peters said. “The people have plenty of money and the letters are lovely and you just have no way of knowing if they had a fight once with the board member’s sister in the elevator of Saks.”