Absolutely -- this is called coppicing. Coppicing is often used on saplings and small pole timber that have been damaged to the point that the tree is no longer capable of producing a quality sawlog or veneer log. Damage may be from environmental (ice, storm, or fire damage) or biological factors (rubbed by deer).
Coppicing utilizes the hardwood tree's established root system to send up multiple basal sprouts from the root collar zone. The intentional act of coppicing should occur as low to the ground as possible -- just barely above the soil line.
