Julie Cohn was fully vaccinated when she went to cheer at her 12-year-old son’s basketball game in March. Midway through the match, he asked to remove his mask because he was getting so hot. She thought little of it.

Three days later, he had a sore throat. He tested positive for the coronavirus, as did his younger brother a few days later. Ms. Cohn cared for them, thinking she was protected, but she woke up with what seemed like a head cold on March 28. The next day, she, too, tested positive.

No vaccine provides perfect protection, and so-called breakthrough infections after coronavirus vaccination are rare and unlikely to lead to serious illness. Federal health officials have told fully vaccinated people they no longer need to wear masks or maintain social distance because they are protected, nor do they need to be tested or quarantine after an exposure, unless they develop symptoms.

Now, the Centers for Disease Control and Prevention has stopped investigating breakthrough infections among fully vaccinated people unless they become so sick that they are hospitalized or die.

Earlier this year, the agency was monitoring all cases. Through the end of April, when some 101 million Americans had been vaccinated, the C.D.C. had received 10,262 reports of breakthrough infections from 46 states and territories, a number that was very likely “a substantial undercount,” according to a C.D.C. report issued on Tuesday.

Genomic sequencing could be done on only 555, or about 5 percent, of the reported breakthrough cases. Over half of them involved so-called variants of concern, including the B.1.1.7 and B.1.351 variants.

Some 995 people were known to have been hospitalized, and 160 had died, though not always because of Covid-19, the new study said. The median age of those who died was 82.

The numbers suggest that the vaccines are highly effective and generally working as expected. On May 1, the agency decided to investigate only the most severe breakthrough infection cases, while still collecting voluntary reports on breakthrough cases from state and local health departments.

The agency will carry out vaccine effectiveness studies that include data on breakthrough cases, but only in limited populations, such as health care workers and essential workers, older adults, and residents at long-term care facilities, a spokeswoman said.

Some scientists support the decision to focus on the illnesses that cause deaths, tax hospitals and overwhelm the health care system. Still, the move has been controversial.

Critics say the agency is missing important opportunities to learn about the real-world effectiveness of the different vaccines and to gather information that might help identify trends in the pandemic’s trajectory — for example, how long vaccine protection lasts, or how various vaccines compare in preventing infection with variants, or whether certain patients like older people are more susceptible to breakthrough infections.

“We are driving blind, and we will miss a lot of signals,” said Ali Mokdad, an epidemiologist at the University of Washington who spent many years as a senior scientist at the C.D.C.

“The C.D.C. is a surveillance agency,” Dr. Mokdad said. “How can you do surveillance and pick one number and not look at the whole?”

The change was announced quietly in a statement on the agency’s website this month. It said the switch “will help maximize the quality of the data collected on cases of greatest clinical and public health importance.”

Asked to explain the change, a C.D.C. spokeswoman said that no vaccine was 100 percent effective, but that the number of Covid-19 cases in fully vaccinated people was small and that no significant demographic trends had been identified.

The change means the agency will continue to investigate cases like the death of a vaccinated resident of a nursing home in Kentucky but not the infections of more than 20 other vaccinated residents and employees in the same home who did not require hospitalization.

It is not clear whether the agency will investigate the outbreak that infected eight vaccinated members of the Yankees organization, because most of them did not develop Covid-19 symptoms at all. (Gleyber Torres, the Yankees’ two-time All-Star shortstop, tested positive after vaccination, as did three coaches and four staff members.)

“We have to prioritize what we’re doing, and the priority is to understand the cases associated with severe disease,” Dr. Kathryn Edwards, a professor of pediatrics at Vanderbilt University School of Medicine who studies vaccine safety and effectiveness, said of the new C.D.C. policy.

Some private health care systems are also studying vaccination failures among employees that will yield useful data, she said, though such studies are not necessarily reflective of broader trends.

“It’s such a rare phenomenon, and it doesn’t change the trajectory of the pandemic,” said Dr. Amesh A. Adalja, a senior scholar at Johns Hopkins Center for Health Security. As long as the disease is not causing people to require hospitalization, “there’s marginal value to tracking it,” he said, adding, “It doesn’t have the ability to crush the hospital system anymore.”

But even relatively mild cases of Covid-19 can lead to persistent long-term health problems, and it will be difficult to know the full scope without tracking mild infections as well.

Ms. Cohn wasn’t hospitalized, but she experienced body aches, chills and digestive problems for about two weeks. She was left with fatigue, headaches, brain fog and vertigo so severe and sudden that she crashed her car into the garage one day.

She is still not back to her daily three-mile runs with her dog because of shortness of breath. “I’m young, 43, healthy, with no pre-existing conditions, but you often find me now on the couch resting,” Ms. Cohn said.

“Don’t people want to know about this?” she asked. “Where do people like me go? What happens next? The practitioners in my life have been shocked and are trying to figure out how to move forward, but there are so many questions. And if no one is studying this, there won’t be answers.”

Another rationale given for not tracking all breakthrough infections is that they are not likely to result in further spread of the virus. But the scientific evidence for this is not conclusive, some experts say.

At Rockefeller University, which routinely tests students and employees for the coronavirus on its campus in New York City, breakthrough infections were identified in two women who had been fully vaccinated and had developed robust immune responses after inoculation, according to a study published in The New England Journal of Medicine.

Both of the vaccinated women, a 51-year-old and a 65-year-old, developed mild symptoms of Covid-19; viral sequencing revealed they had been infected with variants. “One of the individuals had an extraordinarily high viral load,” said Dr. Robert B. Darnell, an investigator at the Howard Hughes Medical Institute and the paper’s senior author.

The patient is not known to have spread the disease to anyone else, he said. Still, he said, “she had twice the transmissible viral load number in a pinhead drop of saliva.”

Diana Berrent, founder of Survivor Corps, a group of people who have had Covid-19, has called for establishing a national registry of all people with Covid-19, including those who had mild and asymptomatic cases, in order to gather as much data as possible for future research.

Many scientists agree: Collecting more data is always preferable to collecting less. “The virus is constantly changing, and we need to stay three steps ahead of it,” said Michael Kinch, an immunologist and associate vice chancellor of the Centers for Research Innovation in Biotechnology at Washington University in St. Louis.

“What if a variant arises that is less responsive or, Lord forbid, unresponsive to the vaccines?” he said. “The way you stop it is good old-fashioned epidemiology, which the C.D.C. has historically done very well. But if you don’t see it coming, you can’t stop it.”