For a compact connected set belonging to some \(o\)-minimal structure (e.g. a semialgebraic or subanalytic set) the author defined the notion of scalar curvature measure which shares many of the properties of the standard scalar curvature of Riemannian manifolds. Carrying on previous work [\textit{A. Bernig}, Adv. Geom. 2, 29-55 (2002; Zbl 1027.53041)] the author relates the scalar curvature measure to curvature bounds in the sense of metric differential geometry and proves here the following result. For a compact connected definable pseudo-manifold \(S\) with curvature bounded from above, the singular part of the scalar curvature measure is non-positive. More precisely, if the dimension of \(S\) equals \(m\) and the curvature bound is \(\kappa\), then \(\text{scal}(S,-)\leq\kappa m(m-1)\text{vol}(S,-)\).