Such infections are called "breakthroughs." They appear to be quite rare and usually cause little, if any, sickness.
These breakthroughs are causing fear that the vaccines are not working.
But most experts disagree.
Dr. Anthony Fauci is the top infectious disease official in the United States. He said infections among a very small number of vaccinated people do not mean the treatment is "failing." He said the vaccines are working well even against the Delta variant. It is now the leading cause of new infections worldwide.
The main reason most experts consider the vaccines successful is that very few breakthrough cases result in hospital treatment or death. Most do not cause any sign of sickness at all.
People who are vaccinated but still get sick generally have mild symptoms for a couple of days before the body starts fighting the virus.
Only about 5,000 out of nearly 160 million vaccinated people have either died or been hospitalized in the U.S. and also tested positive for COVID-19.
Medical experts are following the breakthrough cases. U.S. health agencies are considering whether vaccinated people should get an extra vaccine treatment called a "booster."
People worry when they hear news like that which came from British Prime Minister Boris Johnson. He was very sick with COVID-19 last year and is fully vaccinated now. However, he went into quarantine this week after being around a person who tested positive.
But health officials in the U.S. want to be sure people are not overreacting to the breakthrough cases. That is because many people say they do not want to get the shots. The doctors worry those people will think getting the shot is not worth it if some people still get sick.
Dr. William Schaffner is an infectious disease expert at Vanderbilt University. He said that is the wrong idea. People need to know the vaccines are doing their job.
"The vaccines were developed to keep us out of those terrible institutions we call hospitals," he said. "We have to keep coming back to that."
I'm Dan Friedell.