Some machines reported zero votes for him. In a county with the ability to vote for a straight-party ticket, one candidate's zero votes was a near statistical impossibility. Something had gone quite wrong.

Lee Snover, the chairwoman of the county Republicans, said her anxiety began to pick up at 9:30 p.m. on Nov. 5. She had trouble getting someone from the election office on the phone. When she eventually got through, she said: "I'm coming down there and you better let me in."

With clearly faulty results in at least the judge's election, officials began counting the paper backup ballots generated by the same machines. The paper ballots showed Mr. Kassis winning narrowly, 26,142 to 25,137, over his opponent, the Republican Victor Scomillio.

"People were questioning, and even I questioned, that if some of the numbers are wrong, how do we know that there aren't mistakes with anything else?" said Matthew Munsey, the chairman of the Northampton County Democrats, who, along with Ms. Snover, was among the observers as county officials worked through the night to count the paper ballots by hand.