To explain this, I shall use the same example of a ball bouncing on the ground. In a perfectly ideal world, the ball will never be at rest throughout the bounce. There will be a time $t$ when the ball is going downwards. At time $t+dt$, the ball will be going upwards. This assumes that the coefficient of elasticity of the ball is exactly $1$ and the ball and ground are extremely rigid.
However this cannot happen in real life as there is no such body with perfectly elasticity. Also, the above case would also imply that the force applied by the ground on the ball would be ${\infty}$.
Practically speaking, the ball hits the ground and gets deformed. The velocity slowly decreases to $0$ as the kinetic energy gets used up in changing the shape. The ball will be at rest at a particular moment before bouncing up again.