Government officials, assisted by outside contractors and the National Security Agency, have repeatedly scanned the network and taken some systems offline. But investigators still see signs of the hackers on State Department computers, the people familiar with the matter said. Each time investigators find a hacker tool and block it, these people said, the intruders tweak it slightly to attempt to sneak past defenses.

It isn’t clear how much data the hackers have taken, the people said. They reaffirmed what the State Department said in November: that the hackers appear to have access only to unclassified email. Still, unclassified material can contain sensitive intelligence.

The episode illustrates the two-way nature of high-technology sleuthing. For all of the U.S. government’s prowess at getting into people’s computers through the NSA and the military’s Cyber Command, the government faces challenges keeping hackers out of its own networks. The discrepancy points to a commonly cited problem with defending computers: Playing offense almost is always easier than playing defense.