For 3-dimensional hyperbolic cone structures with cone angles \theta, local rigidity is known for 0 \leq \theta \leq 2\pi, but global rigidity is known only for 0 \leq \theta \leq \pi. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most \pi do not degenerate in deformations decreasing cone angles to zero. In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than 2\pi. These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies of a certain polyhedron. For this construction, we explicitly describe the isometry types on such a hyperbolic polyhedron.