(CNN) -- The federal government messed up following the FBI's investigation of Boston bombing suspect Tamerlan Tsarnaev, Republican Sen. Lindsey Graham said Sunday.
The agency was looking into Tsarnaev's extremist ties two years ago.
"The ball was dropped in one of two ways," Graham said on CNN's "State of the Union."
In 2011, the FBI interviewed Tsarnaev after Russian authorities alerted the U.S. government, according to a senior U.S. official.
Tsarnaev, 26, died early Friday after a shootout with police. His younger brother, Dzhokhar Tsarnaev -- also suspected in the bombings and other crimes in Boston last week -- is hospitalized under federal custody and could be arraigned at his bedside Sunday.
The senator from South Carolina said the FBI either "missed a lot of things" in its investigation, or federal laws did not allow the FBI to "follow up in a sound, solid way."
Tsarnaev, who's ethnically Chechen, traveled to Russia for six months in 2012. Some, including House Homeland Security Chairman Michael McCaul, have argued that Tsarnaev was already being radicalized and received further training during that trip.
Graham said the elder Tsarnaev should have been flagged after the FBI interview and tracked during his visit overseas.
"It's people like this that you don't want to let out of your sight, and this was a mistake. I don't know if our laws were inefficient or if the FBI failed, but we're at war with radical Islamists and we need to up our game," the senator said.
Sen. Chuck Schumer, D-New York, agreed there needs to be an examination of why Tsarnaev's activity was not closely monitored in such a way to prevent last week's terror attack that left three people dead and more than 170 wounded. Authorities also believe the brothers were involved in the shooting death of an MIT police officer Thursday.
"While the FBI has done a very good job over the last 10 years, I think there are questions that have to be answered," Schumer said.
"Why wasn't he interviewed when he came back?"
Responding to a report that Tsarnaev took other trips in addition to his travels in 2012, McCaul told CNN that last year's trip is the only one "that's been made public at this point in time."
"Obviously my biggest concern are other trips," he said. "I could speculate that he was probably going over there more often. We don't have - I don't have the evidence of that just yet. But that's obviously an area of focus. I know the intelligence community is scrubbing through all their records and that travel is very important."
CNN's Lindy Royce-Bartlet contributed to this report.
™ & © 2013 Cable News Network, Inc., a Time Warner Company. All rights reserved.