Psychometric tests, traditionally used to assess humans, are now being applied to Large Language Models (LLMs) to evaluate their behavioral traits. However, existing studies follow a context-free approach, answering each question in isolation to avoid contextual influence. We term this the Disney World test, an artificial setting that ignores real-world applications, where conversational history shapes responses. To bridge this gap, we propose the first Context-Aware Personality Evaluation (CAPE) framework for LLMs, incorporating prior conversational interactions. To thoroughly analyze the influence of context, we introduce novel metrics to quantify the consistency of LLM responses, a fundamental trait in human behavior. Our exhaustive experiments on 7 LLMs reveal that conversational history enhances response consistency via in-context learning but also induces personality shifts, with GPT-3.5-Turbo and GPT-4-Turbo exhibiting extreme deviations. While GPT models are robust to question ordering, Gemini-1.5-Flash and Llama-8B display significant sensitivity. Moreover, GPT models response stem from their intrinsic personality traits as well as prior interactions, whereas Gemini-1.5-Flash and Llama--8B heavily depend on prior interactions. Finally, applying our framework to Role Playing Agents (RPAs) shows context-dependent personality shifts improve response consistency and better align with human judgments. Our code and datasets are publicly available at: https://github.com/jivnesh/CAPE