A graph G is said to be redundantly rigid in d-dimensions if G-e is rigid for every edge e of G. Redundantly rigid graphs are well-known and play an important role in the combinatorial characterizations of several graph classes defined by rigidity and global rigidity properties. One may also consider graphs which remain rigid (or globally rigid) after the removal of any subset of edges (or vertices) of size less than k, for a given integer k>1. We shall present new results concerning highly redundantly rigid graphs, focusing on the three-dimensional versions.