  In this contribution we obtain partial $C^{0,\alpha}$-regularity for bounded
solutions of a certain class of cross-diffusion systems, which are strongly
coupled, degenerate quasilinear parabolic systems. Under slightly more
restrictive assumptions, we obtain partial $C^{1,\alpha}$-regularity. The
cross-diffusion systems that we consider have a formal gradient flow structure,
in the sense that they are formally identical to the gradient flow of a convex
entropy functional. Furthermore, we assume that the cross-diffusion systems are
not volume-filling. The main novel tool that we introduce in this contribution
is a "glued entropy density," which allows us to emulate the classical theory
of partial H\"{o}lder regularity for nonlinear parabolic systems by Giaquinta
and Struwe within this new setting. To demonstrate the applicability of our
results, we give two examples of well-studied cross-diffusion systems that
satisfy our assumptions --one of which is the two component
Shigesada-Kawasaki-Teramoto (SKT) model for population dynamics.
