Cancer patients are increasingly turning to large language models (LLMs) for medical information, making it critical to assess how well these models handle complex, personalized questions. However, current medical benchmarks focus on medical exams or consumer-searched questions and do not evaluate LLMs on real patient questions with patient details. In this paper, we first have three hematology-oncology physicians evaluate cancer-related questions drawn from real patients. While LLM responses are generally accurate, the models frequently fail to recognize or address false presuppositions in the questions, posing risks to safe medical decision-making. To study this limitation systematically, we introduce Cancer-Myth, an expert-verified adversarial dataset of 585 cancer-related questions with false presuppositions. On this benchmark, no frontier LLM -- including GPT-5, Gemini-2.5-Pro, and Claude-4-Sonnet -- corrects these false presuppositions more than $43\%$ of the time. To study mitigation strategies, we further construct a 150-question Cancer-Myth-NFP set, in which physicians confirm the absence of false presuppositions. We find typical mitigation strategies, such as adding precautionary prompts with GEPA optimization, can raise accuracy on Cancer-Myth to $80\%$, but at the cost of misidentifying presuppositions in $41\%$ of Cancer-Myth-NFP questions and causing a $10\%$ relative performance drop on other medical benchmarks. These findings highlight a critical gap in the reliability of LLMs, show that prompting alone is not a reliable remedy for false presuppositions, and underscore the need for more robust safeguards in medical AI systems.