A day after Vice President Pence touted a new policy allowing “any American” to be tested for the novel coronavirus with a doctor’s order, some health officials and physicians expressed concern that people with mild symptoms might overwhelm the nation’s still-limited testing capability.

During a White House briefing on Tuesday, Pence announced government testing guidelines were being loosened to emphasize that physician judgment — not other criteria, such as recent travel to China — would be the main driver in deciding who gets tested. The move drew praise from experts who said the previous guidelines from the Centers for Disease Control and Prevention were far too restrictive in the face of a virus that has spread to more than 75 countries and sickened more than 95,000 people.

But other health experts warned the action might inadvertently send the wrong message, prompting a surge in demand for tests from people with mild symptoms who should simply stay home until they recover. They also noted that laboratory capacity for virus testing, while on the rise, is still lagging. Tests that can be done in doctor’s offices don’t exist.

“This means that right when we need to be careful and methodical about this new testing capacity, we may be overwhelmed,” said Lauren Sauer, who runs emergency preparedness for Johns Hopkins Medicine and the university health system.

Sixty public health labs are now running the just-fixed CDC test. In the next several days and weeks, testing capacity is expected to increase as more labs come online and private companies ship thousands of test kits.

Pence’s announcement took many officials at the Department of Health and Human Services and the CDC by surprise, according to an HHS official who spoke on the condition of anonymity to discuss the matter frankly. On Thursday afternoon, the CDC added the new information to its guidelines for evaluating people who may have the disease, known as covid-19.

Steven Hinrichs, chair of the Department of Pathology and Microbiology at the University of Nebraska Medical Center, said if large numbers of people with colds were to seek testing, it could cause delays in emergency rooms for the seriously ill. During the 2001 anthrax scare, he said, people tried to have their pizza boxes tested because they saw white flour on the bottom.

But state epidemiologists, who are on the front lines of managing the outbreak, said they were largely satisfied with the new criteria.

“They’re getting calls from their providers, hundreds a day, saying, ‘I have this person who may have mild illness but a vague possible exposure, can I get [them] tested?’ ” said Jeffrey Engel, executive director of the Council of State and Territorial Epidemiologists. “They just want to know how to manage the numbers coming in.”

James Lawler, an infectious-disease clinician at the University of Nebraska Medical Center, also welcomed the new policy’s emphasis that testing decisions are up to doctors.

“There was an eventual need to open up that diagnostic ability to allow clinical judgment to drive the decision,” he said.

But by Wednesday, Pence’s pledge had become a rallying cry in Kirkland, Wash., where seven residents of the Life Care Center nursing facility have died of the virus. Many families were demanding that all residents and staff be tested.

“Let’s not wait till they’re symptomatic,” said Kevin Connolly, whose 81-year-old father-in-law is a resident of the center. “These are all elderly. A positive test is a death sentence.”

King County physicians thought that testing everyone who asked was not a medically sound rationale.

Susan J. Baumgaertel, an internist at a large Seattle clinic, described seeing a mother on Wednesday who was worried because she had attended a basketball game before her child’s school closed early for spring break. “I just had to calm her down,” said Baumgaertel, who tried to reassure the patient she probably had the common cold and didn’t need to be tested.

“I don’t think it’s a good public message,” she said of Pence’s announcement.

Once test kits become available in King County later this week, she said, “what we are going to see is suddenly the cases are going to go skyrocketing high because we have testing. The minute you see that, Pence’s words will echo, ‘Oh, we should get tested.’ Which is insane, for most people. You are going to see people come out of the woodwork.”

Some people are struggling to get tested even with the looser criteria. A 51-year-old Indiana woman, who spoke on the condition of anonymity, citing privacy concerns, said she developed a persistent cough about a month ago followed by shortness of breath. The woman, who works at an airport car rental agency where she frequently deals with people traveling from abroad, tested negative for flu.

On Wednesday, she went to her family doctor, who wasn’t sure how to request a test for the coronavirus. She spent an hour in the office waiting for a reply from local and state health officials and in the end was told, incorrectly, that she didn’t qualify because she had not recently traveled out of the country and wasn’t hospitalized.

“I’m not sitting here convinced I have it, nothing like that,” she said. “But none of this other stuff has worked — and because I work in an airport, I just feel it’s prudent at this point to test.”

While doctors have always had authority to order tests, the government’s new guidelines emphasize that. “Clinicians should use their judgment to determine if a patient has signs and symptoms compatible with covid-19 and whether the patient should be tested,” the CDC said on its website.

But Corinne Heinen, who works in a Seattle primary-care clinic, said Wednesday that Pence’s suggestion that any person could ask their doctor to get tested runs counter to strict guidance still in effect at University of Washington Medicine. Under those criteria, patients should be tested only under extremely limited circumstances. “Patients should not self-present for testing,” the guidance says in bold letters.

Under those criteria, it is not enough to have a fever higher than 101 degrees Fahrenheit, a new cough and new shortness of breath, if the person is not sick enough to be hospitalized and does not have any high-risk factors. Only people who have traveled to select international hot spots, or have been close to someone who has, or who have spent time at the nursing facility where deaths have occurred are allowed to get a test. So can people with immune systems suppressed for specific reasons. That means that even “pregnant patients and HIV-positive don’t count,” Heinen said.

She said that on Tuesday, a patient arrived at a primary-care clinic who made “deliveries to people’s doors in Kirkland,” the town in which the nursing facility is located. The patient had respiratory symptoms, Heinen said, but the clinic no longer allows chest X-rays under such circumstances because of the risk to others in the clinic. The patient did not meet the medical system’s test criteria, either.

“This person came in quite ill and did door-to-door deliveries and could not get tested,” Heinen said. Despite what the White House said, she added, “there’s no testing available unless the people are in extremis.”

Maria Sacchetti in Kirkland, Wash., contributed to this report.