Attention mechanisms have recently boosted performance on a range of NLP tasks. Because attention layers explicitly weight input components' representations, it is also often assumed that attention can be used to identify information that models found important (e.g., specific contextualized word tokens). We test whether that assumption holds by manipulating attention weights in already-trained text classification models and analyzing the resulting differences in their predictions. While we observe some ways in which higher attention weights correlate with greater impact on model predictions, we also find many ways in which this does not hold, i.e., where gradient-based rankings of attention weights better predict their effects than their magnitudes. We conclude that while attention noisily predicts input components' overall importance to a model, it is by no means a fail-safe indicator.