Problem definition: Limited resources in the judicial system can lead to costly delays, stunted economic development, and even failure to deliver justice. Using the Supreme Court of India as an exemplar for such resource-constrained settings, we apply ideas from service operations to study delay. Specifically, court dynamics constitute a case-management queue, whereby each case may experience multiple service encounters spread across time, but all are necessarily with the same server. Our goal is to elucidate the drivers of congestion, focusing on metrics such as the expected case-disposition time (delay) and expected number of cases awaiting adjudication (pendency), and leverage this understanding to recommend operational interventions. Methodology/results: We employ data-driven calibrated simulations to model the analytically intractable case-management queue. The life cycle of a case comprises two stages: preadmission (before determining its merit for detailed hearings) and postadmission. Our methodology allows us to capture the queueing dynamics in which the judges are shared resources across the two stages. It also permits modeling of holiday capacity, which is flexibly tailored to address any surplus work that spills over from the regular year. We find that the second stage of this judicial queue is overloaded, but holiday capacity creates a perception of stability by steadying performance metrics. Managerial implications: The sources of inefficiency that drive congestion include a misalignment between scheduling guidelines and judicial capacity, coupled with the requirement to schedule hearings in advance. Together, these factors inhibit utilization of shared capacity across the two-stage judicial queue. We demonstrate how interventions that account for these inefficiencies can successfully tackle judicial delay. In particular, scheduling to improve the allocation of time across preadmission and postadmission cases can cut down the expected delay by as much as 65%. Funding: This study is (partially) supported by a Korea University Business School Research Grant. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0530 .