A convicted felon from Colorado lied about his criminal past while trying to buy guns at two upstate New York stores in 2016.

The man failed the instant background checks and was denied the weapons. Law enforcement was notified because lying on a firearms application is a federal offense, but the man fled back to Colorado before the authorities caught up to him.

Francis Neeley, a 30-year veteran of the Federal Bureau of Alcohol, Tobacco, Firearms and Explosives, took the case to federal prosecutors in upstate New York, seeking charges and an arrest warrant. The prosecutors declined the case, citing a Department of Justice policy to pursue only the most extreme instances in which someone lies on a federal firearms application, Mr. Neeley said.

Mr. Neeley also hit a dead end with New York state prosecutors. They said the case was worthy of prosecution, but they didn’t have funds to send investigators to Colorado and bring the man back to New York, Mr. Neeley recalled.

The man was never charged for lying on the gun applications, and Mr. Neeley said law enforcement doesn’t even know why he made two attempts to purchase a gun.

“We at the ATF have limited resources, and how much time are we going to invest in a case that isn’t going to be prosecuted?” Mr. Neeley said. “You just pray that this guy doesn’t walk into a place and kill someone with a firearm he gets on the street.”

Nearly 16,000 people in the United States died last year as a result of gun violence, and 6,000 people have been killed so far this year, according to the nonprofit Gun Violence Archive.

Several high-profile school shootings have refocused attention on guns, and those on all sides of the issue say more must be done to keep weapons out of the hands of felons, fugitives, the mentally infirm, those with domestic violence records and illegal immigrants — all of whom are supposed to be flagged by the National Instant Criminal Background Check System and prevented from buying firearms.

But tens of thousands of those people try to buy guns each year. Some are successful, and few are prosecuted.

Of those tens of thousands of purchase attempts, the ATF referred fewer than 600 to federal prosecutors between 2008 and 2015. U.S. attorney’s offices then whittled that number down further, considering 254 cases, or an average of 32 a year, for possible prosecution, according to the most recently available statistics compiled by the Justice Department’s inspector general.

Given the lack of penalties, analysts said, there is no real downside to a banned buyer trying.

Yet lives could be at stake.

A 2008 study funded by the Department of Justice found that people who are denied guns are 28 percent more likely to be arrested in the five years after failing background checks compared with the previous five years.

“This is an important enough issue that it ought to be addressed,” said Philip Hilder, a former federal prosecutor who is now a private-practice lawyer. “If it saves just one person’s life, it’s worth the enforcement.”

Attorney General Jeff Sessions promised a change after the Valentine’s Day school shooting in Parkland, Florida, in which 17 people were killed. He directed U.S. attorneys to prosecute what are known in the business as “lie and try” cases.

The George W. Bush and Obama administrations pledged to crack down on these cases, but those promises did not result in meaningful action.

Charles Rose, a Stetson University law professor and former federal prosecutor, said U.S. attorneys just don’t view lie-and-try as important because, ultimately, the person is denied a gun and the case boils down to lies on government paperwork. Other cases take priority, he said.

“If we don’t try these cases, it is not going to make a difference, but if we drop the ball on a terrorism or murder charge, there could be real consequences,” Mr. Rose said.

Prosecutions rose slightly in 2003 during the Bush administration, but analysts said that was misleading. The number of prosecutions was so low that a few additional cases increased the percentage significantly. Politifact, a group of editors and reporters who fact-check politicians’ statements, estimates that the Bush administration prosecuted less than 1 percent of gun application denials.

A decade later, an Obama administration task force proposed ramping up lie-and-try prosecution as part of a group of proposals to reduce gun violence in the wake of the 2012 Sandy Hook Elementary School shooting, which left 28 dead, including 20 children. The measure was hailed by Mayors Against Illegal Guns, a coalition of 800 U.S. mayors who had been urging the Obama administration to take a tougher stance on lie-and-try since 2009.

But prosecutions dropped under President Barack Obama, averaging 32 per year during his first seven years in office.

“We have people with criminal records who have attempted to buy firearms, and one would think any attorney general would make it a high priority to come down on these people, but both Republican and Democratic administrations really don’t,” said Robert Cottrol, a gun control historian who teaches law at George Washington University.

Mr. Sessions vows things will be different this time. The Justice Department said firearms prosecutions have increased nearly 15 percent during the first nine months of this year compared with the same period in 2016, although those cases include offenses in addition to lie-and-try.

Still, the department says Mr. Sessions’ directive to prosecutors to try to use whatever charges they can to prosecute criminals could make lie-and-try cases more common.

“Through initiatives like Project Safe Neighborhoods, [Mr. Sessions] has directed the Department and its federal prosecutors to take a comprehensive approach, including partnerships with state, local and tribal law enforcement agencies to aggressively enforce the law, and partnerships with communities and organizations to help prevent crime and promote public safety,” the Justice Department said in a statement released to The Washington Times.

Mr. Sessions’ supporters say if anyone can reverse a decadeslong trend of apathy toward lie-and-try cases, he will.

John Huber, the U.S. attorney for the District of Utah and a longtime ally of Mr. Sessions, said he has stepped up lie-and-try prosecutions as a result of the March directive.

Mr. Huber said last year that he prosecuted 204 federal firearms cases, but only nine were lie-and-try offenses. Last month alone, he indicted six people accused of trying to circumvent gun background checks.

“Leadership counts,” Mr. Huber told The Times. “The attorney general has articulated a clear vision and clear expectations to U.S. attorneys across the country. He expects us to take this very seriously and apply it to our districts.”

Mr. Huber expects lie-and-try will always be a small sliver of his state’s overall gun prosecutions, but they will make a meaningful impact.

“Lie-and-try is not even close to the majority of gun cases, but it is part of a package of tools I will use to serve my district,” he said.

Other districts also appear to be more aggressive in prosecuting lie-and-try cases. On Friday, two people were charged in a Cleveland federal court with making false statements while attempting to buy a firearm.

Former prosecutors are skeptical that this time will be different.

“It’s a politically expedient thing to say,” Mr. Rose said. “It lets politicians say they are doing something without upsetting the apple cart. It costs them nothing, and from a messaging perspective they can get a lot of bang for their buck: It sounds good, won’t upset anyone, and the news cycle moves on. Arguably, that’s what happened in the last three administrations.”

Mr. Rose said that if the attorney general is serious about tackling lie-and-try, he should create a special unit of prosecutors in each U.S. attorney’s office to focus on the cases.

“The attorney general can say we will do this, but he is not on the ground in those districts,” Mr. Rose said. “If he doesn’t give the offices the resources to do it, he’s telling them that it is not important. Just because he says we are going to do it doesn’t mean it’s going to get done because, historically, it never has.”

Mr. Hilder suggested designating specialized state prosecutors to work with U.S. attorneys offices on these cases. That would be less of a strain of government resources and leave the Justice Department to handle bigger, more-complex cases.

“You need to be creative, look out of the box,” Mr. Hilder said. “If we don’t have the resources, we need to think about who we can get to help.”