Remote Patient Monitoring (RPM) systems will play an important role in the future of healthcare. They will be used to monitor chronic conditions, but may also be employed to detect acute medical conditions and generate alarms in real-time. This real-time responsiveness is a critical design criterion for acute condition detection. The data rate of each sensor represents a hard real-time threshold; if an RPM system cannot process incoming data as quickly as it arrives, its perception of a patient's health status will gradually begin to lag behind that patient's actual status. One effective way to address this issue is to select an operating system (OS) that can effectively manage data analysis for the highest priority tasks under all possible CPU load conditions. This paper evaluates the performance of a real-time operating system (RTOS)-based multi-sensor RPM system. The real-time system performance is measured against a hard realtime processing threshold for five simulated sensor inputs with varying priority levels. The results demonstrate that preemptive scheduling, employed by the RTOS, allows an RPM system under heavy processing load to consistently meet the hard real-time threshold requirements for acute condition detection.