We present CELLO (Covariance Estimation and Learning through Likelihood Optimization), an algorithm for predicting the covariances of measurements based on any available informative features. This algorithm is intended to improve the accuracy and reliability of on-line state estimation by providing a principled way to extend the conventional fixed-covariance Gaussian measurement model. We show that in experiments, CELLO learns to predict measurement covariances that agree with empirical covariances obtained by manually annotating sensor regimes. We also show that using the learned covariances during filtering provides substantial quantitative improvement to the overall state estimate.