Unexpected peaks in volumes of attendances at hospital emergency departments (EDs) have been found to affect waiting times, intensity of care and outcomes. We ask whether these effects of ED crowding on patients are caused by poor clinical prioritisation or a quality-quantity trade-off generated by a binding capacity constraint. We study the effects of crowding created by lower-severity patients on the outcomes of approximately 13 million higher-severity patients attending the 140 public EDs in England between April 2016 and March 2017. Our identification approach relies on high-dimensional fixed effects to account for planned capacity. Unexpected demand from low-severity patients has very limited effects on the care provided to higher-severity patients throughout their entire pathway in ED. Detrimental effects of crowding caused by low-severity patients materialise only at very high levels of unexpected demand, suggesting that binding resource constraints impact patient care only when demand greatly exceeds the ED's expectations. These effects are smaller than those caused by crowding induced by higher-severity patients, suggesting an efficient prioritisation of incoming patients in EDs.