Brownian dynamics simulations of a simple model of a polymer gel were performed to investigate the conditions under which gels show nonergodicity, the presence of frozen density fluctuations which do not relax completely, and to explore possible connections between nonergodicity and the existence of density fluctuations of large spatial extent which are observed in many real gels. The gel was modeled by a random two-dimensional network of springs, formed by bond dilution of a regular triangular lattice. Scatterers located at the nodes of the network moved in Brownian motion. Average scattered intensities and time correlation functions of the intensities were calculated for different gel connectivities and spring constants. These well-defined model systems exhibited both large-scale inhomogeneities and nonergodicity, but no unambiguous connection between the two was found. Nevertheless, in some cases scattering behavior very similar to that of real gels was observed.