“I cannot stay on the board and just shrug my shoulders and move on,” Ms. Zankel wrote.

The museum’s board, which included 27 people before the resignations, according to the Cooper Hewitt website, serves in an advisory role. Several trustees had asked Lonnie G. Bunch III, the Smithsonian’s secretary and top official, to reconsider his decision and reinstate Ms. Baumann. Those trustees have argued that they should have been consulted before he made his decision and that their support of Ms. Baumann’s performance should have been considered.

“We all serve on the board because we share the principles of the institution and the values of its leadership,” Mr. Rockwell said in his letter of resignation. “I cannot in good conscience continue on the board given how Caroline’s dismissal was handled.”

The Smithsonian, which has declined to discuss the reasons for Ms. Baumann’s dismissal, said it regretted the resignations.

“We are sorry to hear this,” said Linda St. Thomas, a spokeswoman, “but it is not unexpected as some board members indicated they did not support the decision.”