Red flags would pop up: The district learned about a past brush with law enforcement and a misstatement on his résumé about where he had earned a doctorate. But Des Moines officials moved ahead to hire him. Two outside companies were involved in vetting Dr. Roberts, who told district officials and a state licensing board that he was a United States citizen.

Nobody seemed to realize that he was lying, and that the man seeking to run Iowa’s largest school district was not allowed to work in this country.

For two years, Dr. Roberts led about 30,000 students and 5,000 employees in Des Moines, earning praise as the district showed signs of academic improvement. Dr. Roberts carried out his work even after an immigration judge in Texas ordered him deported last year, popping into classrooms, asking voters to approve more funds for the district, delivering a Juneteenth speech while wearing one of his signature suits.

Dr. Roberts’s secret life burst into view on Sept. 26, a sunny Friday about a month into the school year, when Immigration and Customs Enforcement agents arrested him after he was found hiding near a trailer park. They said he had fled from officers in a school district-owned Jeep, where a loaded handgun was later found.

Since then, Dr. Roberts has resigned, protesters have held marches calling for ICE to free him and he has been charged with a federal felony in connection to the gun and other weapons officials say they found at his home in Des Moines. Federal officials said those guns had been possessed illegally because Dr. Roberts did not have legal status in the country.

The episode came during President Trump’s sweeping federal crackdown on illegal immigration and on diversity programs in workplaces and school systems. It served as fodder for the administration’s claims that the nation’s immigration enforcement has been woefully impotent. And the revelation upended the image many Americans have of who illegal immigrants are and what sorts of jobs they hold.

Above all, it raised a fundamental question: How could someone the government says should be removed from the country have risen to lead the public school system of a large American city?

The New York Times reviewed hundreds of pages of employment documents, court records and public statements about Dr. Roberts; exchanged texts with the detained superintendent through a jailhouse messaging service; and interviewed people who knew and worked with him over more than a quarter-century, including a colleague who received an unexpected FaceTime call from Dr. Roberts’s phone as he was being arrested.

Dr. Roberts was a smart and ambitious educator who sought out tough assignments and earned the respect of students and colleagues, the review found. But it also revealed a pattern of lying and embellishment that helped propel Dr. Roberts into positions of authority, and a series of oversight failures that allowed his ruse to flourish.

Reached by text message at an Iowa jail, Dr. Roberts said his detention “came as a surprise for many reasons,” but he declined to discuss details of his case or immigration status. His lawyer, Alfredo Parrish, declined to answer questions about his client’s work status, past arrests or inconsistencies on his résumé. He said Department of Homeland Security officials “seem to be making an effort to demonize Dr. Roberts.”

“It is a disturbing pattern that will fail,” Mr. Parrish said in a text message, adding that “the public will see that Dr. Roberts is a good man whose main interest was inspiring and educating young people.”

Some Des Moines residents said they were not sure how to square their positive impressions of the superintendent with ICE’s descriptions of him, especially against the backdrop of the Trump administration’s broader deportation campaign.

The allegations of deception did not match the impression that people like Krista McCalley, a parent who had joined a protest in the hours after Dr. Roberts’s arrest, had of him. “To grapple with that,” she said a week later, “is really, really difficult.”

‘Incredible Energy and Tremendous Rapport’

When he was in his 20s, Ian Roberts arrived at Coppin State University, a historically Black college in Baltimore with Division I sports teams, only after track coaches at other schools had passed on him.

The new recruit proved to be a hard worker and a natural leader. Within a couple years, he had blossomed into the first male N.C.A.A. all-American in Coppin State history.

“He was the premier guy on the track team,” said Ian Smith, a college teammate who recalled that he and his peers “marveled at his ability to just be focused.”

As his results improved, Dr. Roberts competed in ever-more-prestigious track meets, representing Guyana at a world championships in Spain and a Pan American Games in Canada. In 2000, he ran in the men’s 800 meters at the Summer Olympics in Australia, where he did not advance past the first round.

Details of Dr. Roberts’s childhood and early adulthood are fuzzy. He has said on official documents that he was born in 1970 in Guyana, though he was listed with a 1973 birth date in the Olympics. In Des Moines, his official bio said he had spent “most of his formative years” in New York City. At Coppin State, the track team’s media guide listed Guyana’s capital as his hometown and said he had attended high school there.

ICE said that Dr. Roberts entered the United States on a visitor’s visa in 1994, when he was in his 20s. He traveled in and out of the country, and was arrested in New York in 1996 on drug charges and in 1998 on a charge of unauthorized use of a vehicle, according to ICE. The outcome of the drug case was unclear, and ICE said the vehicle case was dismissed. Records of those arrests were not publicly available.

The arrests did not prevent Dr. Roberts from receiving a student visa in 1999, which granted him legal status in the United States until 2004. After receiving a bachelor’s degree from Coppin State in 1998, he enrolled at St. John’s University in New York, where he competed on the track team and received a master’s degree in education in 2000. The following year, he applied to become a permanent resident of the United States, but was turned down.

He returned to Baltimore, where he worked as a teacher for five years before becoming a principal — a fast rise in a school system facing deep challenges.