Rationalization is fundamental to human reasoning and learning. NLP models trained to produce rationales along with predictions, called self-rationalization models, have been investigated for their interpretability and utility to end-users. However, the extent to which training with human-written rationales facilitates learning remains an under-explored question. We ask whether training models to self-rationalize can aid in their learning to solve tasks for the right reasons. Specifically, we evaluate how training self-rationalization models with free-text rationales affects robustness to spurious correlations in fine-tuned encoder-decoder and decoder-only models of six different sizes. We evaluate robustness to spurious correlations by measuring performance on 1) manually annotated challenge datasets and 2) subsets of original test sets where reliance on spurious correlations would fail to produce correct answers. We find that while self-rationalization can improve robustness to spurious correlations in low-resource settings, it tends to hurt robustness in higher-resource settings. Furthermore, these effects depend on model family and size, as well as on rationale content. Together, our results suggest that explainability can come at the cost of robustness; thus, appropriate care should be taken when training self-rationalizing models with the goal of creating more trustworthy models.