Convolutional neural network (CNN) based approaches are the state of the art in various computer vision tasks including face recognition. Considerable research effort is currently being directed toward further improving CNNs by focusing on model architectures and training techniques. However, studies systematically exploring the strengths and weaknesses of existing deep models for face recognition are still relatively scarce. In this paper, we try to fill this gap and study the effects of different covariates on the verification performance of four recent CNN models using the Labelled Faces in the Wild dataset. Specifically, we investigate the influence of covariates related to image quality and model characteristics, and analyse their impact on the face verification performance of different deep CNN models. Based on comprehensive and rigorous experimentation, we identify the strengths and weaknesses of the deep learning models, and present key areas for potential future research. Our results indicate that high levels of noise, blur, missing pixels, and brightness have a detrimental effect on the verification performance of all models, whereas the impact of contrast changes and compression artefacts is limited. We find that the descriptor-computation strategy and colour information does not have a significant influence on performance.