Large language models (LLMs) are increasingly being used to protect sensitive user data. However, current LLM-based privacy solutions assume that these models can reliably detect personally identifiable information (PII), particularly named entities. In this paper, we challenge that assumption by revealing systematic failures in LLM-based privacy tasks. Specifically, we show that modern LLMs regularly overlook human names even in short text snippets due to ambiguous contexts, which cause the names to be misinterpreted or mishandled. We propose AMBENCH, a benchmark dataset of seemingly ambiguous human names, leveraging the name regularity bias phenomenon, embedded within concise text snippets along with benign prompt injections. Our experiments on modern LLMs tasked to detect PII as well as specialized tools show that recall of ambiguous names drops by 20--40% compared to more recognizable names. Furthermore, ambiguous human names are four times more likely to be ignored in supposedly privacy-preserving summaries generated by LLMs when benign prompt injections are present. These findings highlight the underexplored risks of relying solely on LLMs to safeguard user privacy and underscore the need for a more systematic investigation into their privacy failure modes.