Large Language Models (LLMs) are increasingly used in healthcare for tasks such as clinical question answering, diagnosis support, and report summarization. Despite their promise, these models remain highly sensitive to subtle prompt perturbations, both lexical and syntactic, posing serious risks in safety-critical clinical applications. In this study, we conduct a systematic sensitivity analysis to evaluate the robustness of both general-purpose (e.g., GPT-3.5, Llama3) and medical-specific LLMs (e.g., ClinicalBERT, BioLlama3, BioBERT) using the MedMCQA benchmark. We categorize perturbations into natural and adversarial types and examine their effect on model consistency, accuracy, and reliability in clinical reasoning tasks. Our findings reveal that medical LLMs are not intrinsically safe. Even minor variations in phrasing can alter clinical advice, and targeted adversarial prompts can provoke harmful outputs. In high-stakes settings like healthcare, such unpredictability is unacceptable-models that change diagnoses due to reworded inputs or hallucinate medications when slightly rephrased cannot be reliably trusted by clinicians. While models tend to show resilience to simple lexical substitutions or paraphrasing, they often break down under syntactic reordering or misleading contextual cues. This fragility is evident across both general-purpose and domain-specific LLMs. Notably, adversarial manipulations can lead to clinically dangerous outputs, such as recommending incorrect dosages or omitting critical findings.