Blind motion deblurring estimates a sharp image from a motion blurred image without the knowledge of the blur kernel. Although significant progress has been made on tackling this problem, existing methods, when applied to highly diverse natural images, are still far from stable. This paper focuses on the robustness of blind motion deblurring methods toward image diversity-a critical problem that has been previously neglected for years. We classify the existing methods into two schemes and analyze their robustness using an image set consisting of 1.2 million natural images. The first scheme is edge-specific, as it relies on the detection and prediction of large-scale step edges. This scheme is sensitive to the diversity of the image edges in natural images. The second scheme is nonedge-specific and explores various image statistics, such as the prior distributions. This scheme is sensitive to statistical variation over different images. Based on the analysis, we address the robustness by proposing a novel nonedge-specific adaptive scheme (NEAS), which features a new prior that is adaptive to the variety of textures in natural images. By comparing the performance of NEAS against the existing methods on a very large image set, we demonstrate its advance beyond the state-of-the-art.