When the White House R.S.V.P.’d “no” this month to the 2017 Kennedy Center Honors — the first time a president and a first lady have backed out of attending — it allowed both the Trumps and the gala’s hosts to breathe sighs of relief.

But since then, the decision has sparked concerns that a precedent has been set for future presidents, allowing them to skip the starry event when politics intrudes, and upending one of the few Washington traditions left for Republicans and Democrats to come together.

Interviews over the last week with Kennedy Center officials and others involved with the awards shed more light on Mr. Trump’s rare decision to cede ground rather than fight his critics — in this case, some of the honorees who planned to shun a reception at the White House to protest the president’s equivocal remarks about white supremacists in Charlottesville, Va.

Advisers to the first lady, Melania Trump, had been in close contact with officials at the Kennedy Center as the furor over Charlottesville unfolded, and both sides were concerned about protecting their interests. The honors are the Kennedy Center’s most important fund-raiser of the year, and there were concerns that some stars might skip the show itself or that Mr. Trump’s presence might prove a drawback at a moment when some corporate leaders were distancing themselves from him.