The maximum asymptotic bias of S-estimates for linear regressions evaluated over neighborhoods is defined by certain characteristics. The function of this paper is to derive lower and upper bounds on the maximum bias of an S-estimate over neighborhoods defined by these characteristics. After an introduction that explains some background and the purpose of the paper, the proofs are presented. After preliminary propositions, very sophisticated, in-depth theorems and lemmas are detailed. There is some discussion of the normal distribution model, with accompanying theorems and remarks. There is a brief list of references.