The Supreme Court seemed poised on Wednesday to reject a legal theory that puts tight limits on lawsuits seeking to hold police officers accountable for using deadly force.

The case arose, an appeals court judge wrote, from a commonplace occurrence. “A routine traffic stop,” the judge wrote, “has again ended in the death of an unarmed Black man.”

The question for the justices was how closely courts should confine their consideration to “the moment of threat” rather than the larger context of the encounter.

There was something like consensus that the theory was focused too narrowly on the seconds preceding a shooting.

“Everybody agrees it’s wrong,” Justice Neil M. Gorsuch said. “What’s the harm of saying that?”

The case started on an April afternoon in 2016, when Ashtian Barnes, 24, was driving on a highway outside Houston in a car his girlfriend had rented. He was on his way to pick up her daughter from day care.

Though Mr. Barnes did not know it, the car’s license plate was linked to unpaid tolls that had been incurred by another driver. Office Roberto Felix Jr. of the Harris County Constable’s Office pulled the car over based on those unpaid tolls.