Dr. Conley and her colleagues at the cancer institute decided to look back at a variety of early-phase clinical trials of drugs that had been abandoned because, on average, they did not help patients. Were there some participants who had been helped — some exceptional responders?

“Yes, they were actually there,” Dr. Conley said. “Ten percent, maybe less, had this response.”

On Sept. 24, the cancer institute announced that it was sending letters to cancer doctors seeking exceptional responders. The researchers are hoping for tumor samples from 300 such patients, Dr. Conley said, and want to find 100 whose tumor samples contain enough tissue for analysis. So far, they have examined the clinical data for three cases that were sent in. Two of them really are exceptional responders, she said. Now the challenge will be to figure out why.

That may not be easy, said Charles Perou, a professor of molecular oncology at the University of North Carolina at Chapel Hill. Sometimes, he said, researchers will see hundreds of mutations in a cancer and none will explain a patient’s response to a drug. “You are left scratching your head,” he said.

In that light, Dr. Perou said, he can certainly see why The New England Journal of Medicine decided to publish Mrs. Silva’s story.

“It is a stunningly interesting example of molecular genetics and drug response and resistance,” he said.

Mrs. Silva’s symptoms started in 2010 with a badly swollen neck and throat. She saw her doctor, who gave her antibiotics, but the condition recurred again and again.

“I went to my doctor and said, ‘This isn’t normal,’ ” she recalled. Finally, Mrs. Silva, who is 58 and lives in Dartmouth, Mass., ended up at the Dana-Farber Cancer Institute in Boston, where she got terrible news. She had anaplastic thyroid cancer. Anaplastic thyroid cancer, said Dr. Jochen Lorch, her oncologist at Dana-Farber, “is one of the worst cancers you can get.”