Large Language Models (LLMs) have demonstrated remarkable performance in code completion. However, the training data used to develop these models often contain a significant amount of buggy code. Yet, it remains unclear to what extent these buggy instances influence LLMs' performance when tackling bug-prone code completion tasks. To fill this gap, this paper presents the first empirical study evaluating the performance of LLMs in completing bug-prone code. Through extensive experiments on 7 LLMs and the Defects4J dataset, we analyze LLMs' accuracy, robustness, and limitations in this challenging context. Our experimental results show that completing bug-prone code is significantly more challenging for LLMs than completing normal code. Notably, in bug-prone tasks, the likelihood of LLMs generating correct code is nearly the same as generating buggy code, and it is substantially lower than in normal code completion tasks (e.g., 12.27% vs. 29.85% for GPT-4). To our surprise, 44.44% of the bugs LLMs make are completely identical to the pre-fix version, indicating that LLMs have been seriously biased by historical bugs when completing code. Additionally, we investigate the effectiveness of existing post-processing techniques and find that while they can improve consistency, they do not significantly reduce error rates in bug-prone code scenarios. Our research highlights the limitations of current LLMs in handling bug-prone code and underscores the need for improved models and post-processing strategies to enhance code completion accuracy in real-world development environments.