Susan Rice's meeting with her Republican Congressional critics Tuesday didn't go well, and there are increasing signs that the fallout from the attack in Benghazi is hurting the Obama Administration, according to a new poll.

The new CNN/Opinion Research poll shows that 54 percent of Americans disapprove of the Obama Administration's handling of the attack, and 40 percent believe its initial statements about Benghazi were deliberately intended to mislead Americans.

While 40 percent isn't close to a majority, it does suggest that a large number of Americans believe the GOP's charges that Rice's comments were part of a cover-up. And the fact that two in five Americans now believe that there was a cover-up in the Obama White House means Republicans aren't exactly tilting at windmills here. Their charges are catching on with a significant number of Americans.

By way of contrast, in 2007, four years after the start of the war in Iraq, a Gallup poll showed 54 percent of Americans thought the Bush Administration had deliberately misled the American people about Iraq having weapons of mass destruction.

A majority now disapprove of the current administration's handling of the attacks in Libya. Before the attacks, around 40 percent of Americans disapproved of the administration's handling of the the situation in Libya.

(That's not the same question, of course, but it does suggest that the sensitive region is becoming increasingly perilous for the administration politically.)

Following a meeting with the United Nations ambassador this morning, top critic Sen. Lindsey Graham (R-S.C.) said he is even more concerned now about the administration's response to the attacks and about Rice's fitness to be Secretary of State.

"Bottom line, I’m more disturbed now than I was before,” Graham said, according to the Post's Ed O'Keefe.

Sen. Kelly Ayotte (R-N.H.) offered a similar take, and Sen. John McCain (R-Ariz.) said that the group that met with Rice is "significantly troubled by many of the answers we got and some that we didn't get."

Rice, for her part, put out a statement to coincide with the meeting that was notably defensive, saying she never meant to mislead the American people when she suggested that the attack in Benghazi was a result of a spontaneous protest sparked by an anti-Islam video.

Republicans have accused the administration of changing the intelligence community's talking points to blame a demonstration rather than a terrorist attack.

"We stressed that neither I nor anyone else in the administration intended to mislead the American people at any stage in this process, and the administration updated Congress and the American people as our assessments evolved," Rice said in the statement.

Republicans -- especially Graham and McCain -- have focused intensely in recent weeks on Rice's role in the response, so much so that President Obama implored them at a presidential press conference to train their fire on him instead.

They have stuck with attacking Rice, and their strategy appears to be paying dividends.