The departures were an embarrassment for the White House, which was responsible for vetting the prospective jurists, at the end of what has otherwise been a year of success on judicial nominations for Mr. Trump. He has rapidly begun reshaping higher levels of the federal bench by appointing deeply conservative judges.

Following a strategy outlined by Donald F. McGahn II, the White House counsel, Mr. Trump and the Republican majority in the Senate moved swiftly to install Justice Neil M. Gorsuch in a vacant Supreme Court seat and then appointed a dozen appeals court judges — a modern record for a president this early in his tenure.

But the energy and attention that Mr. Trump’s legal team spent on filling vacancies in the upper ranks of the judiciary with powerhouse conservatives may have left less time for vetting nominees for the Federal District Court, where ideology is less important.