Read more: Donald Trump-Kim Jong Un talks end 'without agreement

"The Trump administration has overhyped its claims of success with North Korea," said Bruce Klingner, a senior research fellow at the conservative Heritage Foundation and a former CIA deputy division chief. 

Here's a look at how Trump framed the relationship with North Korea, and how independent experts view it:

"There is no longer a nuclear threat from North Korea"

Trump had barely returned from his first meeting with Kim in June when he declared the challenge posed by North Korea had been solved, setting off alarms with allies. 

"Everybody can now feel much safer than the day I took office," the president tweeted. "There is no longer a Nuclear Threat from North Korea."

Trump's assertion has been blasted by Democrats and Republicans as flatly wrong. North Korea has not given up any of its nuclear weapons. Kim has not disclosed the extent of his nuclear program – a first step toward setting the terms of denuclearization.

Just landed - a long trip, but everybody can now feel much safer than the day I took office. There is no longer a Nuclear Threat from North Korea. Meeting with Kim Jong Un was an interesting and very positive experience. North Korea has great potential for the future!

— Donald J. Trump (@realDonaldTrump) 13 June 2018

Pressed by reporters at his first public event with Kim, Trump denied speculation that he walked back the U.S. goal of denuclearizing the Korean peninsula. Before he left for Vietnam, Trump signaled his immediate goal was for Kim to continue the suspension of missile and nuclear tests.  

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Kim has at least 10 to 20 nuclear weapons and the ability to build dozens more, according to the Arms Control Association, a nonpartisan think tank. In late 2017, North Korea tested its Hwasong-15 intercontinental ballistic missile, which theoretically could reach anywhere in the United States.

"There is still a North Korean nuclear weapons threat," said Daryl Kimball, executive director of the Arms Control Association. "The job of reducing and eliminating the North Korean nuclear and missile threat is not nearly done."

Perhaps more challenging for Trump: There’s little indication Kim wants to relinquish the weapons he has now. Intelligence officials within Trump's own administration said this year that Kim views his nuclear arsenal as crucial to survival.

"We continue to assess that North Korea is unlikely to give up all of its nuclear weapons and production capabilities, even as it seeks to negotiate partial denuclearization steps to obtain key U.S. and international concessions," Dan Coats, the national intelligence director, told lawmakers in January. 

Suspended missile testing

Defending his approach with North Korea, Trump has repeatedly pointed out that Kim is no longer testing missiles.

"Now there's no missile testing. There's no rocket testing. There's no nuclear testing," Trump told reporters gathered in the Oval Office last week.

The president has a point. North Korea last tested a missile in late 2017. But the suspension of testing comes with huge caveats, including the fact that the country has put missile flights on pause before, only to resume them when negotiations break down. 

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Kim declared in April – two months before he met with Trump in Singapore for their first summit – that he would suspend nuclear and missile tests. That was a departure from a series of high-profile tests in the months before.

North Korea launched a missile in November 2017 that climbed 2,800 miles – roughly 11 times higher than the International Space Station – and that experts said could theoretically hit Washington, D.C. Months earlier, North Korea said it tested a hydrogen bomb at its Punggye-ri facility about 350 miles northeast of Pyongyang, an event that registered on seismographs around the planet. 

Kim has explained the latest pause in testing by saying his country's scientists don't need to fire more rockets at the moment. 

"It’s certainly better that the North Koreans have paused their nuclear and ballistic missile testing, but that may only be because North Korea now wants to concentrate on building and producing more long-range ballistic missiles armed with nuclear weapons," Kimball said. 

Almost at war?

At campaign rallies and high-profile speeches, Trump has sought to play up the current U.S. relationship with North Korea by painting a dark picture of where things stood before he became president. To do so, he has claimed President Barack Obama was about to launch a war with North Korea, an assertion that independent experts dispute.

"If I had not been elected," Trump said during his State of the Union address this month, "we would right now, in my opinion, be in a major war with North Korea."

Obama, like his predecessors, looked into the potential fallout from a U.S. strike on North Korea. Obama ultimately decided a military move could lead to a nuclear exchange, killing millions, and he rejected the idea as "unthinkable," according to Bob Woodward’s 2018 book "Fear: Trump in the White House."

Read more: North Korea, the history, the fact and the fiction

Obama embraced a strategy known as “strategic patience,” or the idea of keeping North Korea at arm’s length and hoping international sanctions might weaken Kim’s grip on power. Some criticized the approach policy as doing too little to address the problem, but it was the opposite of preparing for a war. 

It was Trump who, in mid-2017, began rattling the saber with Kim, threatening to bring "fire and fury" on North Korea and touting his bigger "button," a reference to the United States’ larger nuclear arsenal. Kim responded by at one point calling Trump a "dotard."  

"He’s hyping what actually happened," said Harry Kazianis with the Washington-based Center for the National Interest. “I think almost every U.S. administration has weighed military strikes.”

Trump has given North Korea nothing

Trump often rebuffs his critics by saying he has given North Korea nothing in exchange for better relations. But foreign policy experts say the U.S. has delivered to Kim a degree of legitimacy through the joint summits. And they point to the fact that the U.S. military suspended joint training operations with South Korea last year. 

The military exercises, which often involve ships, aircraft and thousands of troops, have been conducted for decades on the Korean Peninsula – sometimes in response to North Korean rhetoric or actions. In 2016, B-52 and B-1B bombers flew in the region after North Korean nuclear tests, according to the Congressional Research Service. 

Some training has continued, but the U.S. has canceled a series of large-scale exercises since the first summit. 

Read more: North Korean leader Kim Jong Un warns Washington not to test Pyongyang's patience

Pentagon officials say the exercises are crucial to maintain the readiness of U.S. and South Korean forces, but the drills have been suspended before – and quickly restarted. The Clinton administration paused the training between 1994 and 1996 in an effort to pave the way for diplomatic talks with North Korea.

When those efforts failed, the exercises resumed.

Trump has described the move as a "cost-saving measure," but neither the White House nor the Pentagon has ever said how much money has been saved. 

"Here we are today, sitting next to each other, and that gives us a hope that we will be successful with time," Kim said during his opening remarks in Vietnam on Wednesday. "And I will really try to make that happen." 

This article originally appeared on USA TODAY: Trump-Kim summit: Fact checking Donald Trump's claims about North Korea