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Reinforcement learning (RL) has been shown to be effective at learning control from experience. However, RL typically requires a large amount of online interaction with the environment. This limits its applicability to real-world settings, such as in robotics, where such interaction is expensive. In this work we investigate ways to minimize online interactions in a target task, by reusing a suboptimal policy we might have access to, for example from training on related prior tasks, or in simulation. To this end, we develop two RL algorithms that can speed up training by using not only the action distributions of teacher policies, but also data collected by such policies on the task at hand. We conduct a thorough experimental study of how to use suboptimal teachers on a challenging robotic manipulation benchmark on vision-based stacking with diverse objects. We compare our methods to offline, online, offline-to-online, and kickstarting RL algorithms. By doing so, we find that training on data from both the teacher and student, enables the best performance for limited data budgets. We examine how to best allocate a limited data budget -- on the target task -- between the teacher and the student policy, and report experiments using varying budgets, two teachers with different degrees of suboptimality, and five stacking tasks that require a diverse set of behaviors. Our analysis, both in simulation and in the real world, shows that our approach is the best across data budgets, while standard offline RL from teacher rollouts is surprisingly effective when enough data is given.