Insisting that attempts at "100 percent security" will always come with inconveniences, President Barack Obama said "we're going to have to make some choices as a society." NBC's Pete Williams reports.

The National Security Agency has at times mistakenly intercepted the private email messages and phone calls of Americans who had no link to terrorism, requiring Justice Department officials to report the errors to a secret national security court and destroy the data, according to two former U.S. intelligence officials.

At least some of the phone calls and emails were pulled from among the hundreds of millions stored by telecommunications companies as part of an NSA surveillance program. James Clapper, the director of national intelligence, Thursday night publicly acknowledged what he called “a sensitive intelligence collection program” after its existence was disclosed by the Guardian newspaper.

Ret. Adm. Dennis Blair, who served as President Obama’s DNI in 2009 and 2010, told NBC News that, in one instance in 2009, analysts entered a phone number into agency computers and “put one digit wrong,” and mined a large volume of information about Americans with no connection to terror. The matter was reported to the Foreign Intelligence Surveillance Court, whose judges required that all the data be destroyed, he said.

Another former senior official, who asked not to be identified, confirmed Blair’s recollection and said the incident created serious problems for the Justice Department, which represents the NSA before the federal judges on the secret court.

The judges “were really upset about this,” said the former official. As a result, Attorney General Eric Holder pledged to the judges that the intelligence agencies would take steps to correct the problem as a condition of renewing the NSA’s surveillance program.

The Justice Department publicly confirmed to the New York Times in April 2009 that Holder had taken “comprehensive steps” to correct a problem in NSA collection after it “detected issues that raised concerns.” But department officials declined to discuss details about what was described at the time as the “over-collection” of information.

In another instance that was made public in July 2012, a U.S. intelligence official acknowledged in a letter to Democratic Sen. Ron Wyden of Oregon that “on at least one occasion” the national security court found that “some collection” by the intellligence community “was unreasonable under the Fourth Amendment” to the U.S. Constitution. The official also wrote that the Office of the Director of National Intelligence believed that the government’s collection of information “has sometimes circumvented the spirit of the law” and that “on at least one occasion” the national security court had “reached this same conclusion.”

Blair declined to say how many times the NSA had had to report the improper collection of information to the court, but indicated it had happened more than once. A spokesman for current DNI Clapper declined comment.

The 2009 incident that Blair described may shed light on an exchange between Clapper and Sen. Ron Wyden, D.-Ore., at a March hearing of the Senate Intelligence Committee.

Asked by Wyden, “Does the NSA collect any type of data at all on millions or hundreds of millions of Americans?” Clapper replied, “Not wittingly. There are cases where they could, inadvertently perhaps collect-but not wittingly.”

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Blair drew a distinction between the “collection” or mining of data on specific U.S. citizens by NSA and the massive trove of phone call information that was turned over to the NSA under a negotiated agreement among intelligence officials, the telecommunications companies and the FISA judges. The purpose of the FISA order was to store information in the event that U.S. intelligence agencies need to access it after getting specific intelligence that somebody in the U.S. might be tied to terrorism. It is only at that point, he explained, that the NSA goes back to the court to get permission to mine or “collect” the data.

But the intelligence community’s distinction between “storing” and “collecting” data does not satisfy privacy and civil liberties advocates. “They are playing games,” said Cindy Cohn, a lawyer for the Electronic Frontier Foundation, which is suing U.S. phone companies over their cooperation with the NSA. Of the improper collection acknowledged by Blair, she said, “Who knows how many times this has happened?”

Meanwhile, a newly empowered federal civil liberties board is asking for a classified briefing "as soon as possible' on U.S. intelligence surveillance programs that have collected hundreds of millions of phone records as well as emails and other content from Internet service providers.

David Medine, chairman of the new Privacy and Civil Liberties Oversight Board, asked for the briefing in a letter Friday to Clapper, the director of national intelligence.

“We want to look into this and find out directly from the source what this is all about and then draw our own conclusions," Medine told NBC News.

The five-member board was authorized by Congress in 2007 to monitor privacy and civil liberties throughout the U.S. government. But after lengthy delays by President Obama in nominating members -- and disputes over its membership -- the panel only officially came into existence last week when Medine was sworn in. It has no website, has yet to move into its office in downtown Washington and has a staff of only two members assigned from other agencies, Medine said.

But Medine noted that the board does have subpoena power over the private sector and the authority to direct U.S. government agency chiefs to provide it with the information it wants.

In his letter to Clapper, Medine noted that his security clearance was still "in process" but asked the intelligence director to give him a "provisional clearance .. in order for me to have authorized access to any classified information conveyed during the briefing."

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