Uncertainty estimation aims to evaluate the confidence of a trained deep neural network. However, existing uncertainty estimation approaches rely on low-dimensional distributional assumptions and thus suffer from the high dimensionality of latent features. Existing approaches tend to focus on uncertainty on discrete classification probabilities, which leads to poor generalizability to uncertainty estimation for other tasks. Moreover, most of the literature requires seeing the out-of-distribution (OOD) data in the training for better estimation of uncertainty, which limits the uncertainty estimation performance in practice because the OOD data are typically unseen. To overcome these limitations, we propose a new framework using data-adaptive high-dimensional hypothesis testing for uncertainty estimation, which leverages the statistical properties of the feature representations. Our method directly operates on latent representations and thus does not require retraining the feature encoder under a modified objective. The test statistic relaxes the feature distribution assumptions to high dimensionality, and it is more discriminative to uncertainties in the latent representations. We demonstrate that encoding features with Bayesian neural networks can enhance testing performance and lead to more accurate uncertainty estimation. We further introduce a family-wise testing procedure to determine the optimal threshold of OOD detection, which minimizes the false discovery rate (FDR). Extensive experiments validate the satisfactory performance of our framework on uncertainty estimation and task-specific prediction over a variety of competitors. The experiments on the OOD detection task also show satisfactory performance of our method when the OOD data are unseen in the training. Codes are available at https://github.com/HKU-MedAI/bnn_uncertainty.