We propose the complete cubic network structure to extend the existing class of hierarchical cubic networks, and establish a general connectivity result which states that the surviving graph of a complete cubic network, when a linear number of vertices are removed, consists of a large (connected) component and a number of smaller components which altogether contain a limited number of vertices. As applications, we characterize several fault-tolerance properties for the complete cubic network, including its restricted connectivity, i.e., the size of a minimum vertex cut such that the degree of every vertex in the surviving graph has a guaranteed lower bound; its cyclic vertex-connectivity, i.e., the size of a minimum vertex cut such that at least two components in the surviving graph contain a cycle; its component connectivity, i.e., the size of a minimum vertex cut whose removal leads to a certain number of components in its surviving graph; and its conditional diagnosability, i.e., the maximum number of faulty vertices that can be detected via a self-diagnostic process, in terms of the common Comparison Diagnosis model.