China and Russia have pressed an informal political and economic alliance against the West. Now they are stepping up the cooperation between their militaries with increasingly provocative joint war games.

Chinese and Russian long-range bombers patrolled together near Alaska for the first time last month. Days earlier, the countries held live-fire naval drills in the hotly contested South China Sea for the first time in eight years. And they have more frequently buzzed the skies and sailed the waters together near Taiwan, Japan and South Korea, where America has strategic interests.

The military exercises are, in some ways, the most vivid expression of an alignment between China’s top leader, Xi Jinping, and President Vladimir V. Putin of Russia as they have sought to challenge their chief geopolitical rival, the United States.

China has been frustrated by American trade restrictions and Washington’s building of security alliances in Asia. It has pushed back by trying to court European countries with trade and building its influence among poorer countries with investments. But those efforts can go only so far in countering the dominance of the United States.

“Beijing increasingly feels that diplomatic and economic actions are not enough to get its points across to Washington, so it is relying more on its military as a tool for signaling. Partnering with Russia is a way to amplify Beijing’s messaging,” said Brian Hart, a fellow with the China Power Project at the Center for Strategic and International Studies.

To Washington, the exercises sow doubts about whether the United States could prevail in a war in Asia against the combined forces of China and Russia. While American war planners have long considered scenarios with China and Russia individually, they have paid less attention to the prospect of the two nuclear-armed states fighting together because it had long seemed so unlikely.