(CNN) - The revelation that al Qaeda leader Osama bin Laden hid in "plain sight" at a house in a bastion of Pakistani military power has touched a nerve in the United States, which relies heavily on the Islamabad government in its grinding war against terrorism.

Many Obama administration officials, lawmakers and observers from across the political spectrum want to know how bin Laden - based at a compound in the military garrison city of Abbottabad - could have eluded Pakistani capture, or whether the government or elements of it harbored bin Laden.

They want to know whether Pakistan is firmly backing the fight against terror or is supportive of militants fighting against troops in Afghanistan.

The United States has a "complicated but important relationship" with Pakistan, White House press secretary Jay Carney said Tuesday.

Carney told reporters that U.S. officials need to learn more about the "support network" that sustained bin Laden in Pakistan. But he also warned against "tarring" everyone in Pakistan's government because of the revelation that bin Laden had been living so close to Islamabad.

There has also been "a great deal of important cooperation" in the fight against Islamic extremism, he said. "The idea that these kinds of complications exist is not new."

But Rep. Peter King, the New York Republican who is chairman of the Committee on Homeland Security, said "this can't be allowed to go on."

In comments Tuesday at a House subcommittee hearing on Pakistan, King noted that bin Laden's compound was near a headquarters for the Pakistani intelligence services.

"There are two possibilities and one answer," King said. "One is that it was a direct facilitation by elements of the Pakistani government, or Pakistani intelligence is entirely inept, and that has not proven to be the case over the years."