What those attacks had in common was the theft of millions of pieces of valuable personal data — including Social Security numbers — that have never shown up on black markets, where such information can fetch a high price. That could be an indicator of state sponsorship, according to James A. Lewis, a cybersecurity expert at the Center for Strategic and International Studies.

But federal investigators, who like other officials would not speak on the record about a continuing inquiry, said the exact affiliation between the hackers and the Chinese government was not fully understood. Their tools and techniques, though, were easily identifiable to intelligence analysts and the security researchers who have been analyzing the breaches at the insurers and the Office of Personnel Management. Federal officials believe several groups were involved, though some security experts only detected one.

“Since mid-2014, we have observed a threat group target valuable ‘personally identifiable information’ from multiple organizations in the health care insurance and travel industries,” said Mike Oppenheim, the manager of threat intelligence at FireEye, a cybersecurity company. “We believe this group is behind the O.P.M. breach and have tracked this group’s activities since early 2013.”

But he argued that “unlike other actors operating from China who conduct industrial espionage, take intellectual property or steal defense technology, this group has primarily targeted information that would enable it to build a database of Americans, with a likely focus on diplomats, intelligence operatives and those with business in China.”

While Mr. Obama publicly named North Korea as the country that attacked Sony Pictures Entertainment last year, he and his aides have described the Chinese hackers in the government records case only to members of Congress in classified hearings. Blaming the Chinese in public could affect cooperation on limiting the Iranian nuclear program and tensions with China’s Asian neighbors. But the subject is bound to come up this week when senior Chinese officials meet in Washington for an annual strategic and economic dialogue.

Though their targets have changed over time, the hackers’ digital fingerprints stayed much the same. That allowed analysts at the National Security Agency and the F.B.I. to periodically catch glimpses of their movements as they breached an ever more diverse array of computer networks.

Yet there is no indication that the personnel office realized that it had become a Chinese target for almost a year. Donna K. Seymour, the chief information officer, said the agency put together last year “a very progressive, proactive plan that allowed us to see the adversarial activity,” and argued that “had we not been on that path, we may never have seen anything” this spring. She cautioned, “There is no one security tool that is a panacea.”