For 15 years, Vladimir V. Putin has confounded American presidents as they tried to figure him out, only to misjudge him time and again. He has defied their assumptions and rebuffed their efforts at friendship. He has argued with them, lectured them, misled them, accused them, kept them waiting, kept them guessing, betrayed them and felt betrayed by them.

Each of the three presidents tried in his own way to forge a historic if elusive new relationship with Russia, only to find their efforts torpedoed by the wiry martial arts master and former K.G.B. colonel. They imagined him to be something he was not or assumed they could manage a man who refuses to be managed. They saw him through their own lens, believing he viewed Russia’s interests as they thought he should. And they underestimated his deep sense of grievance…

Looking back now, aides to all three presidents offer roughly similar takes: Their man was hardly naïve about Mr. Putin and saw him for what he was, but felt there was little choice other than to try to establish a better relationship. It may be that some of their policies hurt the chances of that by fueling Mr. Putin’s discontent, whether it was NATO expansion, the Iraq war or the Libya war, but in the end, they said, they were dealing with a Russian leader fundamentally at odds with the West…

Some specialists said Mr. Obama and his two predecessors saw what they wanted to see. “The West has focused on the notion that Putin is a pragmatic realist who will cooperate with us whenever there are sufficient common interests,” said James M. Goldgeier, dean of international studies at American University. “We let that belief overshadow his stated goal of revising a post-Cold War settlement in which Moscow lost control over significant territory and watched as the West expanded its domain.”