Background: Bug reports are essential to the software development life cycle. They help developers track and resolve issues, but are often difficult to process due to their complexity, which can delay resolution and affect software quality. Aims: This study investigates whether large language models (LLMs) can assist developers in automatically decomposing complex bug reports into smaller, self-contained units, making them easier to understand and address. Method: We conducted an empirical study on 127 resolved privacy-related bug reports collected from Apache Jira. We evaluated ChatGPT and DeepSeek using different prompting strategies. We first tested both LLMs with zero-shot prompts, then applied improved prompts with demonstrations (using few-shot prompting) to measure their abilities in bug decomposition. Results: Our findings show that LLMs are capable of decomposing bug reports, but their overall performance still requires further improvement and strongly depends on the quality of the prompts. With zero-shot prompts, both studied LLMs (ChatGPT and DeepSeek) performed poorly. After prompt tuning, ChatGPT's true decomposition rate increased by 140\% and DeepSeek's by 163.64\%. Conclusions: LLMs show potential in helping developers analyze and decompose complex bug reports, but they still need improvement in terms of accuracy and bug understanding.