Sen. Rand Paul’s filibuster didn’t thrill all of his colleagues.

Almost exactly 24 hours after Mr. Paul began his information-seeking filibuster against John O. Brennan, Sens. John McCain and Lindsey Graham took to the Senate floor to denounce his demands and say he was doing a “disservice” to the debate on drones.

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Mr. McCain quoted from a Wall Street Journal editorial: “The country needs more senators who care about liberty, but if Mr. Paul wants to be taken seriously he needs to do more than pull political stunts that fire up impressionable libertarian kids in their college dorms. He needs to know what he’s talking about.”

The senator went on to say that he didn’t “think that what happened yesterday was helpful to the American people.”

And where Democrats praised Mr. Paul for using Senate rules properly to launch a filibuster, Mr. McCain said it was an abuse of rules that could hurt the GOP in the long run.”What we saw yesterday is going to give ammunition to those who say the rules of the Senate are being abused,” the Arizona Republican said.

Mr. Paul said he was filibustering to get the administration to affirm it won’t kill non-combatant Americans in the U.S. — and his effort was joined by more than a dozen other senators who said they, too, supported his effort to get answers.

Mr. Graham said asking whether the president has the power to kill Americans here at home is a ludicrous question.

“I do not believe that question deserves an answer,” Mr. Graham said.

Mr. Graham and Mr. McCain led a Republican delegation that held a private dinner with President Obama on Wednesday, as Mr. Paul was holding the floor with help from other GOP colleagues.

Mr. McCain even joked about Mr. Graham’s “behavior” at the dinner.

“He was on his best manners and everyone was impressed,” Mr. McCain said.

Speaking to reporters after he came off the floor Wednesday, Mr. Graham said he defends Mr. Paul’s right to ask questions and seek answers, but said the filibuster has actually pushed him to now support Mr. Brennan.

Mr. Graham said he had been inclined to oppose the nomination because he’d found Brennan to be qualified for the job but also “arrogant, kind of a bit shifty.” He said he wasn’t going to filibuster him but would have voted against him on final passage, but now he’ll vote for him.

“I am going to vote for Brennan now because it’s become a referendum on the drone program,” he said.

Mr. Graham, a South Carolina Republican who is up for re-election next year and could face a primary challenge in the deeply conservative state, also fired back against anyone who would question his decision to meet with Mr. Obama.

“If I can’t go have dinner with the president of the United States to talk about the problems that face our nation, I shouldn’t be running,” Mr. Graham said. “If you want to elect me and for me to promise you I’ll never talk to any Democrats or to the president about solving our problems, you’re voting for the wrong guy.”