Real-time schedulability theory requires a priori knowledge of the worst-case execution time (WCET) of every task in the system. Fundamental to the calculation of WCET is a scheduling policy that determines priorities among tasks. Such policies can be non-preemptive or preemptive. While the former reduces analysis complexity and overhead in implementation, the latter provides increased flexibility in terms of schedulability for higher utilizations of arbitrary task sets. In practice, tasks often have non-preemptive regions but are otherwise scheduled preemptively. To bound the WCET of tasks, architectural features have to be considered in the context of a scheduling scheme. In particular, preemption affects caches, which can be modeled by bounding the cache-related preemption delay (CRPD) of a task. In this paper, we propose a framework that provides safe and tight bounds of the data-cache related preemption delay (D-CRPD), the WCET and the worst-case response times, not just for homogeneous tasks under fully preemptive or fully non-preemptive systems, but for tasks with a non-preemptive region. By retaining the option of preemption where legal, task sets become schedulable that might otherwise not be. Yet, by requiring a region within a task to be non-preemptive, correctness is ensured in terms of arbitration of access to shared resources. Experimental results confirm an increase in schedulability of a task set with nonpreemptive regions over an equivalent task set where only those tasks with non-preemptive regions are scheduled nonpreemptively altogether. Quantitative results further indicate that D-CRPD bounds and response-time bounds comparable to task sets with fully non-preemptive tasks can be retained in the presence of short non-preemptive regions. To the best of our knowledge, this is the first framework that performs D-CRPD calculations in a system for tasks with a non-preemptive region.