Is it always possible to determine the position of a body at a given moment in time

Is it always possible to determine the position of a body at a given moment in time, knowing the initial position of this body and the path traveled over a period of time.

The position of the body along the path traveled for a certain period of time can be found only if the body moves along a predetermined trajectory in a given direction. If you know only the dependence of the path on time and the starting point, then you can only be sure that the body has not gone beyond the area bounded by a circle whose radius is equal to the distance traveled.