Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The extra connectivity and the extra edge-connectivity are two important parameters for the reliability evaluation of interconnection networks. The${\mbi {n}}$-dimensional bijective connection network (in brief, BC network) includes several well known network models, such as, hypercubes, Möbius cubes, crossed cubes, and twisted cubes. In this paper, we explore the extra connectivity and the extra edge-connectivity of BC networks, and discuss the structure of BC networks with many faults. We obtain a sharp lower bound of${{g}}$-extra edge-connectivity of an${\mbi {n}}$-dimensional BC network for${{n}} \geq 4$and$1 \leq { {g}} \leq {2^{[{{{n}} \over 2}]}}$. We also obtain a sharp lower bound of${ {g}}$-extra connectivity of an${{n}}$-dimensional BC network for${{n}} \geq 4$and$1 \leq { {g}} \leq 2{ {n}}$which improves the result in [“Reliability evaluation of BC networks,” IEEE Trans. Computers, DOI: 10.1109/tc.2012.106.] for$1 \leq { {g}} \leq { {n}} - 3$. Furthermore, we give a remark about exploring the${ {g}}$-extra edge-connectivity of BC networks for the more general${\mbi {g}}$, and we also characterize the structure of BC networks with many faulty nodes or links. As an application, we obtain several results on the${\mbi {g}}$-extra (edge-) connectivity and the structure of faulty networks on hypercubes, Möbius cubes, crossed cubes, and twisted cubes.