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Since the concept of block matrices stems from a variety of practical applications (discretization of partial differential equations with the finite element method or finite difference methods, structured networks, intermediate transformations in numerical computations etc.) an important task is to analyse their pseudospectra. It is well known that for a fixed perturbation size, the union of pseudospectra of diagonal blocks underestimates the pseudospectrum of a full matrix. So, one is interested to provide an upper estimate by adequately changing the size of the perturbation for the pseudospectra of diagonal blocks. For the case of 2-by-2 block triangular matrix, this was optimally done in Trefethen and Embree (2005). Here we extend this result to general block matrices, and, in turn, obtain some estimates for the distance to instability of a block matrix. © 2020