Caging grasps limit the mobility of an object to a bounded component of configuration space. We introduce a notion of partial cage quality based on maximal clearance of an escaping path. As this is a computationally demanding task even in a two-dimensional scenario, we propose a deep learning approach. We design two convolutional neural networks and construct a pipeline for real-time partial cage quality estimation directly from 2D images of object models and planar caging tools. One neural network, CageMaskNN, is used to identify caging tool locations that can support partial cages, while a second network that we call CageClearanceNN is trained to predict the quality of those configurations. A dataset of 3811 images of objects and more than 19 million caging tool configurations is used to train and evaluate these networks on previously unseen objects and caging tool configurations. Furthermore, the networks are trained jointly on configurations for both 3 and 4 caging tool configurations whose shape varies along a 1-parameter family of increasing elongation. In experiments, we study how the networks' performance depends on the size of the training dataset, as well as how to efficiently deal with unevenly distributed training data. In further analysis, we show that the evaluation pipeline can approximately identify connected regions of successful caging tool placements and we evaluate the continuity of the cage quality score evaluation along caging tool trajectories. Experiments show that evaluation of a given configuration on a GeForce GTX 1080 GPU takes less than 6 ms.