As a network topology for a massively parallel computer system, Generalized Hierarchical Completely- Connected Networks (for short, HCC), which include conventional hierarchical networks, have been proposed. To apply an HCC to a parallel computer system effectively and to execute data processings on the HCC efficiently, the inherent fault-tolerant properties in the HCC must be revealed. However, the properties have not been clarified enough. In this paper, the properties, such as node-connectivity, node-disjoint paths or container, and wide/fault diameter, are verified for the HCC. Especially, the concept of the block-connectivity related to the node-connectivity of HCC is introduced, and the fault-tolerance of HCC is discussed. Furthermore, to clear those properties of HCC, the unknown properties of other hierarchical networks are also cleared. As a result, the HCC has a fault-diameter that is about 1.5 times as long as the diameter of HCC.