Donald Trump is his own worst enemy, as his many ill-advised tweets on the weekend about Michael Flynn, the FBI and Robert Mueller’s Russia probe demonstrate. But that doesn’t mean that Mr. Mueller and the Federal Bureau of Investigation deserve a pass about their motives and methods, as new information raises troubling questions.

The Washington Post and the New York Times reported Saturday that a lead FBI investigator on the Mueller probe, Peter Strzok, was demoted this summer after it was discovered he’d sent anti-Trump...