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Non-volatile memory, such as resistive RAM (RRAM), is an emerging energy-efficient storage, especially for low-power machine learning models on the edge. It is reported, however, that the bit error rate of RRAMs can be up to 3.3% in the ultra low-power setting, which might be crucial for many use cases. Binary neural networks (BNNs), a resource efficient variant of neural networks (NNs), can tolerate a certain percentage of errors without a loss in accuracy and demand lower resources in computation and storage. The bit error tolerance (BET) in BNNs can be achieved by flipping the weight signs during training, as proposed by Hirtzlin et al., but their method has a significant drawback, especially for fully connected neural networks (FCNN): The FCNNs overfit to the error rate used in training, which leads to low accuracy under lower error rates. In addition, the underlying principles of BET are not investigated. In this work, we improve the training for BET of BNNs and aim to explain this property. We propose straight-through gradient approximation to improve the weight-sign-flip training, by which BNNs adapt less to the bit error rates. To explain the achieved robustness, we define a metric that aims to measure BET without fault injection. We evaluate the metric and find that it correlates with accuracy over error rate for all FCNNs tested. Finally, we explore the influence of a novel regularizer that optimizes with respect to this metric, with the aim of providing a configurable trade-off in accuracy and BET.