In practical media distribution systems, visual content often undergoes multiple stages of quality degradations along the delivery chain between the source and destination. By contrast, current image quality assessment (IQA) models are typically validated on image databases with a single distortion stage. In this work, we construct two large-scale image databases that are composed of more than 2 million images undergoing multiple stages of distortions and examine how state-of-the-art IQA algorithms behave over distortion stages. Our results suggest that the performance of existing IQA models degrades rapidly with distortion stages, especially when the distortion types of different stages vary. We also find that full-reference and no-reference frameworks, though both readily applicable, have major drawbacks at predicting the quality of images at middle distortion stages. However, when the quality level of the previous stage is accessible, significantly improved quality prediction performance may be achieved. This study points out a new avenue of degraded-reference IQA research that is both practically desirable and technically challenging.