Recent years have witnessed a growing interest in personalizing the responses of large language models (LLMs). While existing evaluations primarily focus on whether a response aligns with a user's preferences, we argue that factuality is an equally important yet often overlooked dimension. In the context of personalization, we define a model as robust if its responses are both factually accurate and align with the user preferences. To assess this, we introduce PERG, a scalable framework for evaluating robustness in LLMs, along with a new dataset, PERGData. We evaluate fourteen models from five different model families using different prompting methods. Our findings show that current LLMs struggle with robust personalization: even the strongest models (GPT-4.1, LLaMA3-70B) fail to maintain correctness in 5% of previously successful cases without personalization, while smaller models (e.g., 7B-scale) can fail more than 20% of the time. Further analysis reveals that robustness is significantly affected by the nature of the query and the type of user preference. To mitigate these failures, we propose Pref-Aligner, a two-stage approach that improves robustness by an average of 25% across models. Our work highlights critical gaps in current evaluation practices and introduces tools and metrics to support more reliable, user-aligned LLM deployments.