Representatives Duncan Hunter and Chris Collins campaigned this fall while out on bail for felony charges. Representative Greg Gianforte had been convicted of misdemeanor assault. Senator Bob Menendez’s trial on bribery and fraud charges had resulted in a hung jury.

How did voters respond? All four were re-elected last month, Mr. Menendez by 10 percentage points.

Senator Cindy Hyde-Smith of Mississippi wasn’t burdened with legal problems in her runoff election last week, but she did face an uproar after saying she would attend “a public hanging” if a key supporter asked her to — a controversial comment in a state that holds the historical record for the highest number of lynchings. Ms. Hyde-Smith won, too.

Now, as dozens of Democrats consider running for president, the recent success of candidates with varying degrees of baggage has revived interest in a question that has absorbed politicians and strategists since President Trump’s surprise victory two years ago: What, if anything, matters?

Could the plagiarism allegation that Joseph R. Biden Jr. faced in 1988 still be a problem? Mr. Biden dismissed complaints that he lifted passages for his speeches as “much ado about nothing,” but ended his presidential campaign six days later. What about “T-Bone,” a figure in Cory Booker’s stump speeches in 2007, who the senator from New Jersey was accused of making up? At the time Mr. Booker insisted T-Bone was “1,000 percent a real person,” but has never mentioned him again.