The processes that affect sediment quality in drainage systems show high dynamics and complexity. However, relatively little information is available on the influence of both catchment characteristics and meteorological conditions on sediment chemical properties, as those issues have not been widely explored in research studies. This paper reports the results of investigations into the content of selected heavy metals (Ni, Mn, Co, Zn, Cu, Pb, and Fe) and polycyclic aromatic hydrocarbons (PAHs) in sediments from the stormwater drainage systems of four catchments located in the city of Kielce, Poland. The influence of selected physico-geographical catchment characteristics and atmospheric conditions on pollutant concentrations in the sediments was also analyzed. Based on the results obtained, statistical models for forecasting the quality of stormwater sediments were developed using artificial neural networks (multilayer perceptron neural networks). The analyses showed varied impacts of catchment characteristics and atmospheric conditions on the chemical composition of sediments. The concentration of heavy metals in sediments was far more affected by catchment characteristics (land use, length of the drainage system) than atmospheric conditions. Conversely, the content of PAHs in sediments was predominantly affected by atmospheric conditions prevailing in the catchment. The multilayer perceptron models developed for this study had satisfactory predictive abilities; the mean absolute error of the forecast (Ni, Mn, Zn, Cu, and Pb) did not exceed 21%. Hence, the models show great potential, as they could be applied to, for example, spatial planning for which environmental aspects (i.e., sediment quality in the stormwater drainage systems) are accounted.