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A new generation of automated bin picking systems using deep learning is evolving to support increasing demand for e-commerce. To accommodate a wide variety of products, many automated systems include multiple gripper types and/or tool changers. However, for some objects, sequential grasp failures are common: when a computed grasp fails to lift and remove the object, the bin is often left unchanged; as the sensor input is consistent, the system retries the same grasp over and over, resulting in a significant reduction in mean successful picks per hour (MPPH). Based on an empirical study of sequential failures, we characterize a class of "sequential failure objects" (SFOs) -- objects prone to sequential failures based on a novel taxonomy. We then propose three non-Markov picking policies that incorporate memory of past failures to modify subsequent actions. Simulation experiments on SFO models and the EGAD dataset suggest that the non-Markov policies significantly outperform the Markov policy in terms of the sequential failure rate and MPPH. In physical experiments on 50 heaps of 12 SFOs the most effective Non-Markov policy increased MPPH over the Dex-Net Markov policy by 107%.