Is it possible to twist your knee?

Definitely . Knees are flexible joints that can straighten fully and bend until the heel touches the back of the thigh. Knees also rotate a bit side to side. If the force of any movement causes your knee to go beyond the normal limit of motion, you injure theknee. Specifically, the ligaments connecting the thigh bone (femur) and the leg bones (tibia & fibula) are sprained or "twisted" (mild, moderate, severe).
Sure. It's not just possible, it's easy to do. All you have to do is step wrong, get out of a chair awkwardly, or slip.