Recent literature on robust statistical inference suggests that promising outlier rejection schemes can be based on accounting explicitly for sparse gross errors in the modeling, and then relying on compressed sensing ideas to perform the outlier detection. In this paper, we consider two models for recovering a sparse signal from noisy measurements, possibly also contaminated with outliers. The models considered here are a linear regression model, and its natural one-bit counterpart where measurements are additionally quantized to a single bit. Our contributions can be summarized as follows: We start by providing conditions for identification and the Cramér-Rao Lower Bounds (CRLBs) for these two models. Then, focusing on the one-bit model, we derive conditions for consistency of the associated Maximum Likelihood estimator, and show the performance of relevant l1-based relaxation strategies by comparing against the theoretical CRLB.