In hospitals and smart nursing homes, ambient-intelligent care rooms are equipped with many sensors. They can monitor environmental and body parameters, and detect wearable devices of patients and nurses. Hence, they continuously produce data streams. This offers the opportunity to collect, integrate and interpret this data in a context-aware manner, with a focus on reactivity and autonomy. However, doing this in real time on huge data streams is a challenging task. In this context, cascading reasoning is an emerging research approach that exploits the trade-off between reasoning complexity and data velocity by constructing a processing hierarchy of reasoners. Therefore, a cascading reasoning framework is proposed in this paper. A generic architecture is presented allowing to create a pipeline of reasoning components hosted locally, in the edge of the network, and in the cloud. The architecture is implemented on a pervasive health use case, where medically diagnosed patients are constantly monitored, and alarming situations can be detected and reacted upon in a context-aware manner. A performance evaluation shows that the total system latency is mostly lower than 5 s, allowing for responsive intervention by a nurse in alarming situations. Using the evaluation results, the benefits of cascading reasoning for healthcare are analyzed.