“What type of applicants does the FBI investigate?
The FBI conducts background investigations on all persons who apply for employment with the FBI. Additionally, the FBI conducts background investigations for certain other government entities, such as the White House, Department of Justice, Administrative Office of the U.S. Courts, and certain House and Senate committees.”
My understanding is that Obama sat on a Senate Intelligence Committee. If that is correct, then the FBI would have conducted a background check (based on the sensitive intelligence and information that the Senators on that committee handle) on him. Hard to imagine that they would have missed the whole “not a US Citizen” thing.