Recent works have focused on compressing pre-trained language models (PLMs) like BERT where the major focus has been to improve the compressed model performance for downstream tasks. However, there has been no study in analyzing the impact of compression on the generalizability and robustness of these models. Towards this end, we study two popular model compression techniques including knowledge distillation and pruning and show that compressed models are significantly less robust than their PLM counterparts on adversarial test sets although they obtain similar performance on in-distribution development sets for a task. Further analysis indicates that the compressed models overfit on the easy samples and generalize poorly on the hard ones. We further leverage this observation to develop a regularization strategy for model compression based on sample uncertainty. Experimental results on several natural language understanding tasks demonstrate our mitigation framework to improve both the adversarial generalization as well as in-distribution task performance of the compressed models.