LETTERKENNY ARMY DEPOT, Chambersburg, Pennsylvania—Linda Woodford spent the last 15 years of her career inserting phony numbers in the U.S. Department of Defense's accounts.

Every month until she retired in 2011, she says, the day came when the Navy would start dumping numbers on the Cleveland, Ohio, office of the Defense Finance and Accounting Service, the Pentagon's main accounting agency. Using the data they received, Woodford and her fellow DFAS accountants there set about preparing monthly reports to square the Navy's books with the U.S. Treasury's—a balancing-the-checkbook maneuver required of all the military services and other Pentagon agencies.