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Test-time training (TTT) through self-supervised learning (SSL) is an emerging paradigm to tackle distributional shifts. Despite encouraging results, it remains unclear when this approach thrives or fails. In this work, we first provide an in-depth look at its limitations and show that TTT can possibly deteriorate, instead of improving, the test-time performance in the presence of severe distribution shifts. To address this issue, we introduce a test-time feature alignment strategy utilizing offline feature summarization and online moment matching, which regularizes adaptation without revisiting training data. We further scale this strategy in the online setting through batch-queue decoupling to enable robust moment estimates even with limited batch size. Given aligned feature distributions, we then shed light on the strong potential of TTT by theoretically analyzing its performance post adaptation. This analysis motivates our use of more informative self-supervision in the form of contrastive learning for visual recognition problems. We empirically demonstrate that our modified version of test-time training, termed TTT++, outperforms state-of-the-art methods by significant margins on several benchmarks. Our result indicates that storing and exploiting extra information, in addition to model parameters, can be a promising direction towards robust test-time adaptation.