That question could ultimately face two cultural institutions: the Detroit Institute of Arts, whose artwork may be sold off because the city has declared bankruptcy, and the South Street Seaport Museum, which is desperately trying to stay afloat.

Museums don’t often go out of business. They cut back, they pare down, but they tend to persevere as cultural anchors of their communities. And neither of these institutions has announced plans to close; both are hoping to weather their current storms.

But both are arguably in jeopardy — the Detroit Institute because a sale would denude its prestigious collection of its most valuable artworks and compromise its integrity, since nonprofit museums, founded in the public trust, are ethically obligated not to sell pieces except to acquire others; the Seaport because it has struggled for years to pay its bills and recently lost its white knight when the Museum of the City of New York said it could no longer afford to run the museum, which was damaged by Hurricane Sandy.

Should these institutions find themselves forced to close, they could look to the Fresno Metropolitan Museum of Art and Science, which shut down in 2010, as an object lesson in the complex, painful process of dissolution.

Although vastly different in scale and reputation from the august Detroit Institute and the endearing Seaport Museum, the Fresno Met had certain key things in common with those institutions. It was a nonprofit whose building was in the end owned by the city. (The City of Detroit owns the Institute and much of its collection, and the Seaport Museum leases its building from the city’s Economic Development Corporation.) Also like the Detroit Institute and the Seaport Museum, the Fresno Met was a flagship attraction and a local point of pride.

“It was extremely difficult,” said Dana Thorpe, the Fresno Met’s last executive director. “For many people in the community, this was their Disneyland.”

The Fresno Met was one of nearly 30 museums that decided to close in 2009, the last year statistics were compiled, according to the American Alliance of Museums, a national association. The alliance no longer tries to calculate the number of closings, said a spokesman, Dewey Blanton, because many go unreported.

The Fresno Met didn’t have a lot of history behind it. It was created in 1984 in the former home of The Fresno Bee newspaper, a 1922 Renaissance Revival building on the National Register that the Bee’s owner donated. Among the largest arts institutions in California’s Central Valley, the museum had a few gems in its collection like American Indian baskets and a cache of Ansel Adams photographs.

For a while, the Fresno Met thrived as a center for programs in art and science. But by the time Ms. Thorpe became director in June 2008, the museum was foundering. A $28 million renovation project was $15 million over budget and three years behind schedule, closing the museum during construction.

Although the Met finally managed to reopen in November 2008, and attendance reached a record 110,000 over the ensuing year, contributions diminished as the recession hit donors’ stock portfolios. In 2009, carrying more than $4 million in debt, the museum cut its operating budget by 45 percent, went through two rounds of layoffs and closed its Chagall exhibition early.

“It was a Taj Mahal and a beautiful museum, but the demographics were not there to support it,” said Stewart Randall, the former board president.

The museum sought help from the City of Fresno, which agreed to guarantee a $15 million loan. But the economic downturn left the museum unable to raise money to refinance the loan, so the city took over the building. And the writing was on the wall.

“We were spending $300,000 a month and had income of $200,000 a month,” Mr. Randall said. “That’s when it became evident to me that there was no way this was going to survive.”

Staff members were let go. The collection had to be sold. Debts had to be paid. “We were now talking about closing the museum with dignity and grace,” Ms. Thorpe said.

To deal with all this, the museum hired Riley C. Walter, a Fresno bankruptcy lawyer.

The museum considered filing for Chapter 11 protection. But, after calculating the potential costs and delays, it instead pursued an insolvency proceeding to benefit creditors that is governed by state law rather than federal bankruptcy law.

The procedure turns the museum’s assets over to an assignee — in this case, O. James Woodward III, a prominent local lawyer and art collector — who then oversees their disposal.

“It was far less expensive and far faster than a Chapter 11,” Mr. Walter said. “In Chapter 11, you would have to go through elaborate notice procedures and give a lot more opportunity for people to object.”

The museum notified creditors of its liquidation, so that they could file claims, then auctioned off non-art items like pedestals and display cases. Sotheby’s handled the sale of the most valuable art works, which brought about $2 million; other auction houses sold the rest.

The unsecured creditors received 80 cents on the dollar.

At the beginning of the process, a group of local patrons lent the museum $675,000 and took a lien against its assets. That money was used in the liquidation for payroll while the museum was obtaining valuations from auction houses. This loan was paid off through the Fresno Met’s first sale of assets.

The liquidation prompted only one lawsuit: The family of Ansel Adams said the photographs were meant to stay in a museum. So the family traded other pieces it had for those in the collection, based on a fair market value price determined by an auction house.

Note to institutions contemplating a similar move: The aftermath wasn’t all grim. A science and math exhibition for children was bought by a children’s museum in north-central California; a large collection of original boxed puzzles went to a toy museum. The museum’s building is now rented out by the city as commercial space. And some of the staff members and trustees landed at the Fresno Art Museum, an older institution that focuses on contemporary American art.

Still, “it was really unfortunate,” Mr. Walter said. “And it’s led to there being one fewer cultural amenity in the whole region.”

Ms. Thorpe said she had received calls for advice from other museum directors whose institutions were in similar straits on how to avoid the same fate.

“I share the story of the Fresno Met,” she said. “I never want to see another museum close, even though I know it continues to happen.”