A pitstop in Moscow?
Graham said the Bush administration telegraphed in small but telling ways that other foreign countries, particularly Iraq, took precedence over the bilateral relationship with Moscow.
In 2006, for example, the White House asked the Kremlin for permission for Bush to make a refueling stop in Moscow on his way to an Asia-Pacific summit meeting. But it made clear that Bush was not looking to meet with Putin, whom he would see on the sidelines of the summit.
After Russian diplomats complained, Graham was sent to Moscow to determine if Putin really wanted a meeting and to make clear that if there was one, it would be substance-free.
In the end, the two presidents met and agreed to ask their underlings to work on a nonproliferation package.
(Read more: Putin admits Crimea involvement; warns on gas)
"When the Russian team came to Washington in December 2006, in a fairly high-level ... group, we didn't have anything to offer," Graham said. "We hadn't had any time to think about it. We were still focused on Iraq."
Graham said that the Bush administration's approach slighted Moscow. "We missed some opportunities in the Bush administration's initial years to put this on a different track," Graham said. "And then later on, some of our actions, intentional or not, sent a clear message to Moscow that we didn't care."
Three train wrecks
Bush's relationship with Putin unraveled in 2008. In February, Kosovo unilaterally declared independence from Serbia with the support of the United States - a step that Russia, a longtime supporter of Serbia, had been trying to block diplomatically for more than a decade. In April, Bush won support at a NATO summit in Bucharest for the construction of a missile defense system in Eastern Europe.
Bush called on NATO to give Ukraine and Georgia a so-called Membership Action Plan, a formal process that would put each on a path toward eventually joining the alliance. France and Germany blocked him and warned that further NATO expansion would spur an aggressive Russian stance when Moscow regained power.
In the end, the alliance simply issued a statement saying the two countries "will become members of NATO." That compromise risked the worst of both worlds - antagonizing Moscow without giving Kiev and Tbilisi a roadmap to join NATO.
The senior U.S. official said these steps amounted to "three train wrecks" from Putin's point of view, exacerbating the Russian leader's sense of victimization. "Doing all three of those things in kind of close proximity - Kosovo independence, missile defense and the NATO expansion decisions - sort of fed his sense of people trying to take advantage of Russia," he said.
In August 2008, Putin struck back. After Georgia launched an offensive to regain control of the breakaway, pro-Russian region of South Ossetia, Putin launched a military operation that expanded Russian control of South Ossetia and a second breakaway area, Abkhazia.
The Bush administration, tied down in Iraq and Afghanistan, publicly protested but declined to intervene militarily in Georgia. Putin emerged as the clear winner and achieved his goal of standing up to the West.
Only one major issue
After his 2008 election victory, Barack Obama carried out a sweeping review of Russia policy. Its primary architect was Michael McFaul, a Stanford University professor and vocal proponent of greater democracy in Russia who took the National Security Council position previously held by Thomas Graham.
In a recent interview, McFaul said that when Obama's new national security team surveyed the administration's primary foreign policy objectives, they found that few involved Russia. Only one directly related to bilateral relations with Moscow: a new nuclear arms reduction treaty.
The result, McFaul said, was that relations with Moscow were seen as important in terms of achieving other foreign policy goals, and not as important in terms of Russia itself.
"So that was our approach," he said.
Obama's new Russia strategy was called "the reset." In July 2009, he traveled to Moscow to start implementing it.
In an interview with the Associated Press a few days before leaving Washington, Obama chided Putin, who had become Russia's prime minister in 2008 after reaching his two-term constitutional limit as president. Obama said the United States was developing a "very good relationship" with the man Putin had anointed as his successor, Dmitry Medvedev, and accused Putin of using "Cold War approaches" to relations with Washington.
"I think Putin has one foot in the old ways of doing business and one foot in the new," Obama said.
In Moscow, Obama spent five hours meeting with Medvedev and only one hour meeting with Putin, who was still widely seen as the country's real power. After their meeting, Putin said U.S.-Russian relations had gone through various stages.
"There were periods when our relations flourished quite a bit and there were also periods of, shall we say, grayish mood between our two countries and of stagnation," he said, as Obama sat a few feet away.
At first, the reset fared well. During Obama's visit, Moscow agreed to greatly expand Washington's ability to ship military supplies to Afghanistan via Russia. In April 2010, the United States and Russia signed a new START treaty, further reducing the U.S. and Russian nuclear arsenals. Later that year, Russia supported sweeping new U.N. economic sanctions on Iran and blocked the sale of sophisticated, Russian-made S-300 anti-aircraft missile systems to Tehran.
Experts said the two-year honeymoon was the result of the Obama administration's engaging Russia on issues where the two countries shared interests, such as reducing nuclear arms, countering terrorism and nonproliferation. The same core issues that sparked tensions during the Bush administration - democracy and Russia's neighbors - largely went unaddressed.