WASHINGTON — When Special Agent Adrian Hawkins of the Federal Bureau of Investigation called the Democratic National Committee in September 2015 to pass along some troubling news about its computer network, he was transferred, naturally, to the help desk. His message was brief, if alarming. At least one computer system belonging to the D.N.C. had been compromised by hackers federal investigators had named “the Dukes,” a cyberespionage team linked to the Russian government. The F.B.I. knew it well: The bureau had spent the last few years trying to kick the Dukes out of the unclassified email systems of the White...