  Zhu et al. [Theoret. Comput. Sci. 758 (2019) 1--8] introduced the $h$-edge
tolerable diagnosability to measure the fault diagnosis capability of a
multiprocessor system with faulty links. This kind of diagnosability is a
generalization of the concept of traditional diagnosability. A graph is called
a maximally connected graph if its minimum degree equals its vertex
connectivity. It is well-known that many irregular networks are maximally
connected graphs and the $h$-edge tolerable diagnosabilities of these networks
are unknown, which is our motivation for research. In this paper, we obtain the
lower bound of the $h$-edge tolerable diagnosability of a $t$-connected graph
and establish the $h$-edge tolerable diagnosability of a maximally connected
graph under the PMC model and the MM$^*$ model, which extends some results in
[IEEE Trans. Comput. 23 (1974) 86--88], [IEEE Trans. Comput. 53 (2004)
1582--1590] and [Theoret. Comput. Sci. 796 (2019) 147--153].
