Cord problems come in two types: electrical and mechanical. Some vacuums have auto-feed cords that are powered with springs or other devices that can break. Electrical problems are more serious, and you should never use a vacuum with a cord that has an electrical short. Cord conductors can break inside the insulated coating of the cord, causing the short.
Shorts often create heat as the conducting material around the short is too thin and frayed to safely provide current at normal temperatures. Because of this, a short can cause a fire. You should have any shorted electrical cord fixed by a qualified electrician or vacuum cleaner service and repairman.