Some states have seen a desperately needed increase in their bandwidth to test for the coronavirus in recent days — but the United States' testing capacity still lags far behind other nations.

"We have had a phenomenal increase in testing," New York Gov. Andrew Cuomo said Monday at a press conference where he announced there were 950 cases of the virus statewide, the highest of any state in the country.

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When the outbreak first came to New York, the state set a target goal of running 1,000 tests a day. By the end of this week, the state expects to be able to perform 7,000 tests per day — an "exponential increase of what we have done," Cuomo said. Along with the uptick in tests, Americans should expect to see a rise in confirmed cases, he added.

"What happens when the testing capacity increases, the number of positives increase by definition," he said.

Identifying who is sick is a key part of mitigating a pandemic. So far, the ability to freely test anyone showing symptoms, which include fever, cough and shortness of breath, has eluded U.S. public health officials.

Many state officials have expressed exasperation over the testing rollout, as it was hampered at first by flawed test kits sent out by the Centers for Disease Control and Prevention to public health laboratories across the country, and then limited by what was then narrow criteria from the agency for who qualified for a test. A shortage of a chemical needed to run the tests last week added to the problems.

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The ability to test has since increased across the country. In Washington state, which reported 796 total cases on Monday, a spokeswoman for the state Department of Public Health told NBC News that more equipment was brought in for testing earlier this month and that capacity continued to increase at the state public health lab, where more than 200 samples can be tested each day, with a goal to test 400 samples a day.

But increases in testing capacity has not necessarily translated into easier access to tests for many Americans. Social media is flooded with horror stories of having symptoms, but being denied a test by hospitals or public health departments. And the nurses' union, National Nurses United, has stated that "countless" health care workers exposed to the coronavirus have been refused a test for it.

Frustrations have grown as other countries have surpassed with ease the number of tests that are being performed in America. In South Korea, a leader in coronavirus testing capabilities, 15,000 people are tested per day.

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The U.S., in contrast, has struggled to test barely double that in total. As of Monday, NBC News estimated that there have been at least 33,000 tests done, based on figures listed on about 40 state public health department websites across the country. (It is not clear whether every state also includes the number of tests run by private laboratories.)

The test kit failures were acknowledged last week by a top public health official.

"The system is not really geared to what we need right now," Dr. Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, testified in a House hearing Thursday. "That is a failing. Let's admit it."

But there is hope: In the past couple of weeks, commercial labs including Quest Diagnostics and LabCorp have started testing for COVID-19, the illness caused by the coronavirus.

Both Quest and LabCorp said in statements to NBC News that they will each have the ability to perform more than 10,000 tests per day by the end of the week and 20,000 tests per day by the end of March. The trade association both the labs belong to, the American Clinical Laboratory Association, said in a statement that "assuming there are no delays or shortages of necessary materials and supplies, commercial capacity is expected to exceed 280,000 tests per week by April 1.”