If you take an aluminum rod and tap one end with a hammer, the disturbance travels along the rod at the speed of sound in aluminum, which is about 5000 m/s. This speed is what determines the frequency of the ringing that you hear. The speed is many orders of magnitude less than c. If it were higher than c for some other substance (one that was very stiff and had a very low density), then it would be possible to use the vibrations to transmit information faster than the speed of light, which is forbidden by relativity; it leads to paradoxes, since there would be frames of reference in which the signal was received earlier than it was transmitted. This tells us that relativity imposes limits on the properties of materials. It isn't surprising that such limits exist, since the properties of materials are determined by the electrical interactions between atoms, and those interactions propagate at no more than c.
A similar example from general relativity is that we can't use a rope to retrieve an object from inside the event horizon of a black hole. If the rope were to be strong enough to support even its own weight, then the speed of sound in the rope would be greater than c, which is impossible.
Although relativity doesn't allow the existence of perfect rigidity as a passive property of a substance, it still allows us to define a notion of rigidity called Born rigidity (Born 1909). In a Born-rigid object, an observer at rest relative to a certain part of the object sees that part of the object as always maintaining a constant distance between neighboring parts. Born rigidity can't be a passive property of a material; to achieve Born rigidity, one has to carry out a pre-planned program of applying forces to different parts of the object as a function of time.
The Herglotz-Noether theorem says that Born rigidity is incompatible with the kinds of free rotations and translations that we expect nonrelativistically to be able to apply to a rigid body. It is not possible for a Born-rigid body to change its angular velocity, and if such a body is rotating, its center of mass can't be accelerated.
Historically, this kind of thing was studied intensively ca. 1910 both because of the desire to resolve paradoxes such as the Ehrenfest paradox and because people were trying to make a theory of electrons as extended objects, in order to avoid the infinite energy inherent in the field of a point charge.
Max Born. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. (The theory of the rigid electron in relativistic kinematics) Annalen der Physik (Leipzig), Annalen der Physik 30, 1; also referred to as 335 (11), 1-56 (vierte folge, band 30), 1909.