John Troy Davis, 44, faces a charge of assault on a federal employee. If convicted, he faces up to 10 years in prison and a $250,000 fine.
The alleged threat happened two days before six people died in Saturday's shootings at a Safeway grocery store in Tucson, Ariz., where U.S. Rep. Gabrielle Giffords was meeting constituents. A federal judge, a congressional aide and a young girl were among the six killed, while Giffords and 13 others were wounded.
Jeff Dorschner of the U.S. attorney's office said there were no indications that the cases were related.
FBI agent Jonathan D. Grusing said in an arrest affidavit that Davis called Bennet's office Thursday to complain about his Social Security benefits. At one point, according to the document, Davis told a Bennet staffer that he is schizophrenic and needs help and that he "may go to terrorism."
Davis appeared Monday before a federal magistrate judge, who formally advised him of the charge and appointed the public defender's office to represent him.
Davis is due back in court Thursday.
While attacks on Congress members are rare, threats are not. Dorschner said federal prosecutors in Colorado file an average of six to eight cases per year involving threats against U.S. representatives and senators.
"The tragic events in Arizona on Saturday confirm that the U.S. attorney's office and our law enforcement partners must continue to treat every threat of violence against elected officials and other federal employees and staff as a matter of utmost seriousness," U.S. Attorney for Colorado John Walsh said in a statement, adding that other threats to public officials are under investigation.
Bennet issued a statement through a spokesman confirming the threat and said his office has full confidence in the FBI and Capitol Police, who arrested Davis.
In two unrelated cases last week, two men were charged with threatening the life of President Barack Obama. Donald Edward Hatten, 47, of Colorado Springs, walked into an FBI office there Thursday and told agents he planned to hitchhike to Pennsylvania, obtain a handgun from his father's house, then hitchhike to the White House to kill Obama, according to an affidavit. He told a Secret Service agent that voices in his head were telling him to do that.
In a separate case, Weld County jail inmate Jesse Arispe was charged Wednesday with sending a threatening letter to Obama in July. Details on that case were not immediate available.
Davis, whose last known address was a motel west of downtown Denver, was well known to staffers in Bennet's office because he had called several times before to complain his Social Security benefits, according to the affidavit.
But during one call Thursday, Grusing quotes Davis as telling a staffer: "I'm just going to come down there and shoot you all." Davis called again and spoke to another staffer, this time saying: "To get your attention, I will go down there and set fire to the perimeter."
Staffers told investigators Davis was upset about not having a hearing about his benefits, although Davis failed to attend a hearing arranged by Bennet's office.
An officer with the FBI's Rocky Mountain Safe Streets Task Force visited Davis' motel and learned that he had been told not to return because "he caused problems and was always angry."