“I guess accidents can happen,” one said in a very brief telephone conversation.

A State Department spokeswoman, Marie Harf, told reporters on Monday, “We aren’t going to discuss, you know, publicly operational details about the possible response options,” adding that “as we implement our responses, some will be seen, some may not be seen.”

There was no definitive way, at least in the short term, to determine whether the connection had been cut, overloaded, or attacked. And security experts cautioned that there could be many reasons for Monday’s failure. North Korea could be pre-emptively taking its systems offline to prepare for an attack, some said.

Chris Nicholson, a spokesman for Akamai, an Internet content delivery company, said it was difficult to pinpoint the origin of the failure, given that the company typically sees only a trickle of Internet connectivity from North Korea. The country has only 1,024 official Internet protocol addresses, though the actual number may be a little higher. That is fewer than many city blocks in New York have. The United States, by comparison, has billions of addresses.

But when the sun rose in North Korea on Tuesday morning, the few connections to the outside world — available only to the elite, the military, and North Korea’s prodigious propaganda machine — were still out.

As the morning wore on, however, some of the connections began to come back after a blackout of nearly 10 hours, though there was still very little traffic, according to CloudFlare, an Internet company in San Francisco.