When Rep. Keith Ellison ran for Democratic National Committee chairman, he faced questions about past associations with the Nation of Islam and Louis Farrakhan. On MSNBC’s “Morning Joe” in December 2016, Mr. Ellison angrily accused his critics of a “smear campaign” for “talking about something that happened in 1995,” when Mr. Ellison was 32. It turns out Mr. Ellison—who lost his bid but is now the DNC’s deputy chairman—wasn’t telling the full story.

In 2006, during his first run for Congress from Minnesota, Mr. Ellison conceded...