Whether snipping with scissors or opening a box, humans can quickly understand the 3D configurations of familiar objects. For novel objects, we can resort to long-form inspection to build intuition. The more we observe the object, the better we get at predicting its 3D state immediately. Existing systems, however, are limited to either optimizing underlying representations from multi-view observations or training a feed-forward predictor from supervised datasets. We introduce Predict-Optimize-Distill (POD), a self-improving framework that interleaves prediction and optimization in a mutually reinforcing cycle to achieve better 4D object understanding with increasing observation time. Given a multi-view object scan and a long-form monocular video of human-object interaction, POD iteratively trains a neural network to predict local part poses from RGB frames, uses this predictor to initialize a global optimization which refines output poses through inverse rendering, then finally distills the results of optimization back into the model by generating synthetic self-labeled training data from novel viewpoints. Each iteration improves both the predictive model and the optimized motion trajectory, creating a virtuous cycle that bootstraps its own training data to learn about the pose configurations of an object. We also introduce a quasi-multiview mining strategy for reducing depth ambiguity by leveraging long video. We evaluate POD on 14 real-world and 5 synthetic objects with various joint types, including revolute and prismatic joints as well as multi-body configurations where parts detach or reattach independently. POD demonstrates significant improvement over a pure optimization baseline which gets stuck in local minima, particularly for longer videos. We also find that POD's performance improves with both video length and successive iterations of the self-improving cycle, highlighting its ability to scale performance with additional observations and compute.