Successful adoption of deep learning (DL) in the wild requires models to be: (1) compact, (2) accurate, and (3) robust to distributional shifts. Unfortunately, efforts towards simultaneously meeting these requirements have mostly been unsuccessful. This raises an important question: Is the inability to create Compact, Accurate, and Robust Deep neural networks (CARDs) fundamental? To answer this question, we perform a large-scale analysis of popular model compression techniques which uncovers several intriguing patterns. Notably, in contrast to traditional pruning approaches (e.g., fine tuning and gradual magnitude pruning), we find that ``lottery ticket-style'' approaches can surprisingly be used to produce CARDs, including binary-weight CARDs. Specifically, we are able to create extremely compact CARDs that, compared to their larger counterparts, have similar test accuracy and matching (or better) robustness---simply by pruning and (optionally) quantizing. Leveraging the compactness of CARDs, we develop a simple domain-adaptive test-time ensembling approach (CARD-Decks) that uses a gating module to dynamically select appropriate CARDs from the CARD-Deck based on their spectral-similarity with test samples. The proposed approach builds a "winning hand'' of CARDs that establishes a new state-of-the-art (on RobustBench) on CIFAR-10-C accuracies (i.e., 96.8% standard and 92.75% robust) and CIFAR-100-C accuracies (80.6% standard and 71.3% robust) with better memory usage than non-compressed baselines (pretrained CARDs and CARD-Decks available at https://github.com/RobustBench/robustbench). Finally, we provide theoretical support for our empirical findings.