HANOI, Vietnam — Over the past quarter-century, three American presidents have tried cajoling, threatening and sabotaging North Korea in its efforts to build a nuclear arsenal. Eventually each turned to negotiations, convinced that an isolated, broken country would surely choose economic benefits for its starving populace over the bomb.

President Trump was the fourth to test that proposition, but with a twist: Engaging in the sort of direct talks that his predecessors shunned, the president traveled 8,000 miles for his second summit meeting in less than a year with Kim Jong-un, the North Korean leader, betting that his self-described skills as a master negotiator would make all the difference.

As it turned out, they didn’t. The meeting in Vietnam ended in shambles on Thursday when Mr. Kim insisted on a full lifting of sanctions, according to Mr. Trump, and would not agree to dismantle enough of his nuclear program to satisfy American demands. The North Koreans later contradicted Mr. Trump, saying they had demanded only a partial lifting of sanctions, but they confirmed that they had offered to dismantle their main nuclear site, at Yongbyon.

The split underscored the risk of leader-to-leader diplomacy: When it fails, there are few places to go, no higher-up to step in and cut a compromise that saves the deal.