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This paper digs into the relationship between grasps and cages of a rigid body, in particular into the use of cages as waypoints to grasp an object. We introduce the concept of pregrasping cages, caging configurations from which the object can be reached while maintaining the cage on it. In the well understood case of two-fingered manipulators, the squeezing/stretching caging characterization implies that all cages are indeed pregrasping cages and, as a consequence, are useful waypoints to grasp an object. We show in this paper that the same does not hold for more than two fingers. There are caging configurations from which a grasp of the object cannot be reached without breaking the cage on it. We explore the natural generalization of the squeezing/stretching characterization to the case of n fingers in arbitrary dimension and exploit it to give sufficient conditions for a cage to be a pregrasping cage.