More than 40 million people — the equivalent of one out of every four American workers — have filed for unemployment benefits since the coronavirus pandemic grabbed hold in mid-March, the government reported on Thursday, an astounding tally that rivals the bleakest years of the Great Depression.The latest additions — the 2.1 million people who filed state unemployment claims last week — may not be only a result of fresh layoffs, but also evidence that states are working their way through some of the choking backlog.“We’re still catching up,” Diane Swonk, chief economist at the accounting firm Grant Thornton, said of the newest claims. “The lags have been long.”The Labor Department report marks the eighth week in a row that new jobless claims, on a seasonally adjusted basis, dipped from the peak of almost 6.9 million — but the level is still far above any other historical highs.At the same time, overcounting in some places and undercounting in others makes it difficult to precisely measure the number of layoffs caused by the pandemic — and in devising a policy response.“When we think about what to do when benefits expire, it would be helpful to know how many people are actually getting them,” said Elizabeth Pancotti, a research assistant at the National Bureau of Economic Research. While the Labor Department reports may be the best source of information, she said, they offer an “incomplete picture.”Shelter-in-place orders and business restrictions have been lifting across the country, and there is evidence in the report that some workers are being called back: The number of people receiving state jobless benefits dropped by roughly 3.8 million to 21.1 million for the week ended May 16.But as Ms. Swonk noted, “it’s not enough to offset the extraordinary economic devastation and job losses associated with Covid-19.”And while rehiring certainly accounts for a chunk of that decline, workers who had exhausted their weekly state benefits would also be reflected.Reopenings remain bumpy and incomplete, and flare-ups of the coronavirus continue to disrupt business. On Tuesday, Ford Motor temporarily halted production at the Kansas City assembly plant in Missouri to deep clean after an employee tested positive for the virus. Two other Ford plants — in Chicago and Dearborn, Mich. — were also temporarily closed.In Thursday’s report, the department offered two sets of figures. One includes the more than 40 million people who have applied for state benefits and is seasonally adjusted. The other includes those who applied under the new federal emergency program called Pandemic Unemployment Assistance and is not seasonally adjusted. So far, more than 10 million have applied.Pandemic Unemployment Assistance, part of an expanded palette of jobless benefits passed by Congress two months ago, is meant to help freelancers, gig workers, the self-employed and others who would not normally qualify under state rules.But several economists suspect that there is a lot of double counting and warn against simply adding figures from the two programs together.Some states, flooded with applicants, were slow to put the pandemic program into effect. Initially, many people were mistakenly told they were ineligible. Others were instructed to apply for state benefits first and be rejected before applying for the federal benefits.That confused application process has caused potentially millions of laid-off workers to be counted twice. States are also weeding out duplicate applications from frustrated filers who had trouble getting through or did not receive any response after weeks of waiting.Then there are the mistakes. A data entry slip-up caused Massachusetts to pump up the number of federal claims by nearly a million last week. The previous week, a similar flub in Connecticut mistakenly inflated its total by a quarter of a million.“It’s unclear if states are including duplicate claims due to error, fraud or the Pandemic Unemployment Assistance program,” said Ernie Tedeschi, a policy economist at Evercore ISI in Washington.But that is not all. That system is also probably missing millions of other laid-off workers.As of Tuesday, three states had not put the pandemic unemployment insurance program into effect, and several others have yet to report any claims. Thirteen states have not started another federal emergency relief program, to provide an additional 13 weeks of benefits to workers who have exhausted their state benefits.Laid-off workers who have not applied for benefits and those who have left the labor force entirely are not included in the claims numbers. Nor are any of the eight million undocumented workers who lost their jobs. They are not eligible for any benefits. Neither are new graduates just entering the labor force.Matthew Wilson, 24, who lost his barista job in Philadelphia, was turned down because he had been working in the state for less than a year.“It doesn’t make any sense — I moved, and now I’m magically not qualified for unemployment?” said Mr. Wilson, who relocated to Pennsylvania after graduating from Tufts University in Massachusetts last year. He appealed the decision and heard last week that his claim had been approved, but he hasn’t received any money. His partner, who also lost her job as a barista, has applied four times but has yet to collect benefits.As for regular unemployment benefits, states draw up their own rules and administer benefits. The result is that in some places, like Florida, Texas and Arizona, only a small fraction of jobless workers are receiving benefits, while other states offer much broader coverage.The way “initial claims” are counted may also vary by state, with some excluding claims that have not been processed.Laurie Yadoff, a lawyer at Coast to Coast Legal Aid of South Florida, said she had about 100 clients who qualify for regular state benefits but have had trouble filing. Many are poor and older, with limited or no access to the internet. “A lot of them fall into the regular state benefit program, and a lot of them are straightforward, and a lot of them are still not getting money,” she said.Even when Ms. Yadoff has been able to get someone on the phone, the person at the other end often doesn’t know the answer. “People are desperate and frustrated,” she said. “They don’t know what to do.”Allison Hester, who is 50 and lives in Little Rock, Ark., applied for unemployment after being laid off from her job in content marketing at a window cleaning supply company in March, but was never able to get through. “I tried on and off for a month, but our system was so overwhelmed.”She returned to her job this month. “It feels good, but I don’t feel secure anymore,” Ms. Hester said. “I don’t take anything for granted.”Nelson D. Schwartz and Tiffany Hsu contributed reporting.