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The ability of a dynamical system to remain stable in the face of perturbations in values of the system's parameters/coefficients is an important safety-attribute in control-system analysis and design and is referred-to as "stability-robustness". The most fundamental question in the study of stability-robustness is to identify, or safely-approximate, the "extent/range" of parameter-variations for which the system remains stable, in some defined sense. In this paper we consider the class of "constant" linear dynamical systems and show that in some, seemingly-normal sub-cases, an asymptotically-stable, constant linear dynamical system can exhibit rather unusual non-robust stability features. An Example is presented and the unique structural-property characterizing the non-robust stability behavior is identified.