The effectiveness of cancer screening and adherence to cancer screening guidelines can be inhibited by long wait times for screening appointments. We develop a queueing network model of screening for a disease within a population of patients with imperfect adherence to screening guidelines to characterize the relationship between the screening request frequency rate and the wait time for screens. We first use our model to derive the capacity needed by a given system or the population size a given system can serve to guarantee a certain service level. We analyze routine screening for average-risk patients as well as the additional capacity required for diagnostic screening and surveillance of high-risk patients. We perform a numerical study using national public health data for colorectal cancer (CRC) screening in the United States and provide insights for public health officials, providers, and policy makers by showing how queueing models can be applied to regional planning decisions for determining CRC screening capacity requirements.