Resistive memory is a promising alternative to SRAM, but is also an inherently unstable device that requires substantial effort to ensure correct read and write operations. To avoid the associated costs in terms of area, time and energy, the present work is concerned with exploring how much noise in memory operations can be tolerated by image classification tasks based on neural networks. We introduce a special noisy operator that mimics the noise in an exemplary resistive memory unit, explore the resilience of convolutional neural networks on the CIFAR-10 classification task, and discuss a couple of countermeasures to improve this resilience.