Can the federal government investigate a case at the same time as state police?
With many potential criminal cases, law enforcement in the state in which the alleged crime occurred does the investigating. After investigating, it hands the case off to the local prosecuting agency. For instance, if there’s been a theft, the local police or sheriff’s department might look into the case and report its findings to the county district attorney’s office. The district attorney’s office would then decide whether to file charges against the suspect. (Prosecutors may also go the route of a grand jury—see Grand Jury Basics.)
But the federal government has jurisdiction over some of the same cases that state governments can prosecute. (See State vs. Federal Prosecution.) That means that the United States can prosecute the defendant even if the state does the same. This ability occasionally leads the federal government to investigate a case simultaneously as, but independently from, a local police agency. Or the federal government and a state government might investigate a case together. (For a real-life instance of a state and the feds investigating at the same time, see Federal Charges for Shooting of Unarmed Missouri Teen?)