Although capacity has improved in recent weeks, supply shortages remain crippling, and many regions are still restricting tests to people who meet specific criteria. Antibody tests, which reveal whether someone has ever been infected with the coronavirus, are just starting to be rolled out, and most have not been vetted by the Food and Drug Administration.

Concerns intensified on Wednesday as Senate Democrats released a $30 billion plan for building up what they called “fast, free testing in every community,” saying they would push to include it in the next pandemic relief package. Business leaders, who participated in the first conference call of Mr. Trump’s advisory council on restarting the economy, warned that it would not rebound until people felt safe to re-emerge, which would require more screening…

State health officials and medical providers around the country say they are unable to test as many people as they would like. Many of them say the biggest challenge is getting not the diagnostic tests themselves but the supplies to process them, including chemical reagents, swabs and pipettes. Manufacturers are facing a huge global demand as every country fights the pandemic, with many attempting the widest-scale testing they have ever undertaken.