It is not clear what the inspector general concluded about the dress, but another focus of its investigation was the location of the wedding, the LongHouse Reserve in East Hampton. The reserve, a 16-acre property with carefully tended gardens and outdoor sculptures, is run by a nonprofit founded by Jack Lenor Larsen, a prominent textile designer who lives on the property and is a friend of Ms. Baumann. (In 2015, the Cooper Hewitt awarded Mr. Larsen one of its annual national design awards.)

Mr. Larsen welcomed Ms. Baumann’s use of the property for the ceremony (the reception was held on Shelter Island) and did not charge her. According to the people familiar with the Smithsonian’s decision, it took issue with the fact that Ms. Baumann allowed Mr. Larsen’s nonprofit to hold its board meetings at no cost in a conference room at the Cooper Hewitt.

“How can it be inappropriate for a nonprofit to use a conference room for a board meeting?” said Matko Tomicic, the executive director of LongHouse. But the Smithsonian was apparently concerned about the appearance that favors had been exchanged and believed that Ms. Baumann should have reported the free use of the reserve as a gift.

The Smithsonian is not a federal agency, but it was created by Congress and is partly funded by the government. In accordance with its charter, its board of regents comprises three members each from the House and Senate, the vice president, the chief justice and members of the public.

In many ways, it behaves like an appendage of the government, and has a conflict of interest policy that mimics those of many public agencies: “Employees must not engage in private or personal activities that might conflict, or appear to conflict, with Smithsonian interests, such as using Smithsonian employment for private gain” or “giving preferential treatment to any person or company for any reason.”

Like many federal departments, it also has an independent inspector general. But Ms. Sleeper said that she was troubled by the behavior of the agents who interviewed her. A few weeks later, she sent an email to the inspector general’s office saying that the agents sounded biased against Ms. Baumann and predisposed to find fault.