<p>The aim of this paper is to find an upper bound for the box-counting dimension of uniform attractors for non-autonomous dynamical systems. Contrary to the results in the literature, we do not require the symbol space to have finite box-counting dimension. Instead, we ask a condition on the semi-continuity of pullback attractors of the system as time goes to infinity. This semi-continuity can be achieved if we suppose the existence of finite-dimensional exponential uniform attractors for the limit symbols, that is, the symbols associated with the asymptotic vector fields of the non-autonomous differential equation. After showing these new results, we apply them to study the box-counting dimension of the uniform attractor for a non-autonomous reaction-diffusion equation, and we find a class of forcing terms for this equation such that their associated symbol spaces have infinite box-counting dimension, but the uniform attractors of the dynamical systems generated by these forcing terms have finite box-counting dimension anyway.</p>