For weeks, immigrant rights groups had searched for the right case to highlight their battle against President Trump’s get-tough immigration policy. They think they’ve found it this week in Juan Manuel Montes-Bojorquez, a Dreamer who went public with his story that he was deported despite having been approved under the Obama administration’s temporary amnesty.

Top Democrats in Washington seized on the case, with House Minority Leader Nancy Pelosi calling Mr. Montes outstanding and saying his deportation was a disgrace to American values. Sen. Richard J. Durbin, Illinois Democrat, demanded explanations, saying Homeland Security Secretary John F. Kelly had personally promised last month that nobody approved under President Obama’s Deferred Action for Childhood Arrivals (DACA) policy would be deported by the Trump administration.

Immigrant rights activists rallied outside a Homeland Security Department office in Washington on Wednesday, demanding that Mr. Montes be reimported from Mexico.

“We will stand up to racist agents that continue to target our community,” United We Dream community organizer Deyanira Aldana said though a bullhorn as she led the protest.

But the case is turning out to be much more complicated, and as Homeland Security officials tell it, Mr. Montes’ situation is not the clear-cut case of a clean-cut kid being ousted for no good reason.

Federal officials say Mr. Montes had been approved for DACA through early next year but broke terms of the program by leaving the U.S. without permission. He was nabbed by Border Patrol agents Feb. 19 while trying to sneak back into the country.

He also had several run-ins with the law, including a shoplifting conviction last summer.

“The advocacy groups were absolutely salivating, smelling blood in the water and ready to attack the administration for being overzealous with enforcement and for having the audacity to deport someone with DACA, whom they believe should be immune from deportation,” said Jessica Vaughan, policy studies director at the Center for Immigration Studies, which pushes for stricter immigration limits. “This case seems to prove the old adage, ‘If it sounds too good to be true, it probably isn’t true.’”

Tensions over Mr. Trump’s immigration policies have been building for months as immigrants anticipated a crackdown.

The president’s tough talk has already dried up the flow of migrants across the southwestern border, with rates at their lowest point in 40 years. But illegal immigrants inside the U.S. have hunkered down and vowed to resist efforts to deport them.

Nowhere has that been more true than for the Dreamers, the young adult illegal immigrants long seen as the most sympathetic figures in the debate. Usually brought to the U.S. as children, many of them don’t even have memories of their home countries.

After efforts to grant them legal status stalled in Congress in 2010, Mr. Obama moved in 2012 to use executive authority to create the DACA program, granting a renewable two-year stay of deportation and work permits to those who had worked toward high school diplomas and kept out of major criminal trouble. More than 750,000 immigrants were approved for the program under Mr. Obama.

The Dreamers have built themselves into a potent political force, earning speaking roles at the Democratic National Convention and pressuring lawmakers for favorable treatment.

Indeed, they have even forced a reversal from Mr. Trump, who during the campaign said he would repeal the DACA program but who now takes a softer line. He said he wants to find a solution that will satisfy Dreamers.

While most other illegal immigrants are once again targets for deportation, Dreamers remain generally out of bounds, administration officials say. DACA recipients snared in immigration raids have even been let go.

Activists, though, had been searching for a case that would show Mr. Trump is breaking his promises to Dreamers.

They thought they found one earlier this year in Daniel Ramirez Medina, a 23-year-old snared in a raid targeting his father, also an illegal immigrant. Immigration agents accused the young man of gang ties, which would invalidate his DACA status, but his attorneys said the evidence was fabricated. He was eventually released, though his case is still making its way through immigration courts.

Activists then seized on the case of another Dreamer, a 22-year-old woman in Mississippi who allowed her DACA status to lapse. But she, too, was released by immigration authorities last month.

This week, the activists zeroed in on Mr. Montes, the 23-year-old who was deported back to Mexico despite having his DACA status renewed last year.

In an interview with USA Today, Mr. Montes said he was first deported Feb. 18 after an agent accosted him and asked him questions, then refused to let him get his wallet where he had proof of his DACA status.

Mr. Montes said he was so frustrated with being sent to Mexico that he tried to sneak back into the U.S. the next day, but he was quickly arrested and deported again.

U.S. Customs and Border Protection, the agency that oversees the Border Patrol, gave a completely different version of events. The agency first said Mr. Montes had fallen out of DACA status in 2015, then recanted and acknowledged that he had been approved through 2018.

But the agency disputed most of the rest of Mr. Montes’ tale. CBP said there was no record of a Feb. 18 encounter and deportation, just the Feb. 19 arrest after Mr. Montes was caught sneaking into the U.S. That meant he had left the country to go to Mexico without getting permission — a breach of DACA.

“During [Mr. Montes’] detention and arrest by the United States Border Patrol on February 19, he admitted to agents that he had illegally entered the United States and was arrested,” the agency said in a statement.

CBP said it couldn’t verify Mr. Montes’ contention that he had been in the U.S. since age 9. The agency said its first encounter with him was in 2010, when he tried to enter without authorization and instead accepted a quick deportation.

It’s unclear how he managed to be approved for DACA in 2014. The terms of DACA require illegal immigrants to have lived in the U.S. continuously since 2007.

CBP said Mr. Montes never mentioned DACA status after he was caught by the Border Patrol. Even if it had come up, the agency said, he violated the terms and was subject to deportation.

Mr. Montes also had several convictions on his record for driving without a license, and a shoplifting conviction from last summer.

Mr. Montes’ defenders dismissed the inconsistencies in his story and said they had no qualms about making him the poster child for their campaign against Mr. Trump’s policies.

“At the end of the day, this is a young man who grew up in the United States since he was 9,” said Julieta Garibay, campaign director for United We Dream. “We know the kind of intimidation tactics that ICE and CBP uses.

“He had DACA. He should not be deported. Period,” said Ms. Garibay.

But Ms. Vaughan said immigrant rights advocates are overselling DACA, treating it as a form of legal status with certain rights, when even the Obama administration said the program was discretionary and approvals could be revoked.

Ms. Vaughan said leaving the country without permission has always been grounds for revoking DACA status.

“The advocacy groups and news reporters should do their homework before making him a poster boy for either DACA or opposition to Trump’s policies,” she said. “They could well have egg on their face after this one.”

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