Suppose Γ is an S-arithmetic subgroup of a connected, semisimple algebraic group G over a global field Q (of any characteristic). It is well-known that Γ acts by isometries on a certain CAT(0) metric space X_S = ∏_{v ∈ S} X_v, where each X_v is either a Euclidean building or a Riemannian symmetric space. For a point ξ on the visual boundary of X_S, we show there exists a horoball based at ξ that is disjoint from some Γ-orbit in X_S if and only if ξ lies on the boundary of a certain type of flat in X_S that we call "Q-good.'' This generalizes a theorem of G.Avramidi and D.W.Morris that characterizes the horospherical limit points for the action of an arithmetic group on its associated symmetric space X.