As muscles contract, they provide forces countering inversion proportional to the intensity of contraction. It is important to recognize that these tendons are very important stabilizers as well. It is only when these muscles and tendons are overpowered or caught off guard that they fail in their mission, and the underlying ligaments are tested. In the ankle, the actual grade of the sprain is almost never of consequence, as virtually all initial first-time sprains are treated the same. Compression with an elastic wrap or sleeve should continue until swelling is gone.
Rest does depend upon the degree of injury.
Some mild sprains may necessitate only a few seconds of rest, such as during a run. Obviously, if one can keep running after a rollover during training or a race, the sprain is quite mild. However, it would be wise to elevate, ice, and wrap immediately afterwards. A good rule of thumb is that when one can stand without pain, try walking. When one can walk without pain, try jogging.