Data published in April are the best evidence yet of how the San Antonio program is outperforming its peers. In a nine-year trial comparing a group of people who took part in Project Quest with a group who did not, the Quest graduates ended up earning $5,000 more annually. That was especially significant since earnings gains from training programs typically fade over time, Mr. Osterman said.

“The results were stunning,” said Mark Elliott, president of the Economic Mobility Corporation, a nonprofit research group that carried out the study. “These are the largest sustained earnings impacts we’ve ever seen in a work-force development program.”

Just as Project Quest is an outlier nationally, so is the United States among Western countries. Other nations have done a better job with training programs, said Anthony Carnevale, director of the Georgetown Center on Education and the Workforce.

Canada has a more robust two-year college system, Mr. Carnevale said, “and they spend a lot less than we do on postsecondary education and training and get a lot better results than we do.”

In Europe, there are separate tracks for apprenticeship programs in high school, but Americans are uncomfortable with tracking students at such a young age.

“You can go to Europe, see apprenticeships are great and come back and say, eureka, I found the answer!” Mr. Carnevale said. “But 20 years later, you’re still going to be banging your head against the wall.”