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Test-Time Adaptation (TTA) aims to quickly adapt a pre-trained Deep Neural Network (DNN) to shifted test data from unseen distributions. Early TTA works only targeted simple and restrictive test scenarios that did not align with the philosophy of TTA that emphasizes practicality. Subsequent research efforts have thus been geared towards exploring more realistic test scenarios. In the same spirit, this work investigates for the first time TTA with data streams contaminated with out-of-distribution (OOD) data. Surprisingly, we observe the existence of benign OOD data that can improve TTA performance. We provide meaningful insights into the causes of benign OOD-contamination by analyzing the feature space of the pre-trained DNN. Inspired by these empirical findings, we propose R3, a novel TTA algorithm that specifically targets OOD-contaminated streams. Our experimental results verify that R3 improves competitive baselines by up to nearly 3%p on OOD-contaminated streams created with CIFAR-10-C and ImageNet-C.