Large Language Models are a promising tool for automated vulnerability detection, thanks to their success in code generation and repair. However, despite widespread adoption, a critical question remains: Are LLMs truly effective at detecting real-world vulnerabilities? Current evaluations, which often assess models on isolated functions or files, ignore the broader execution and data-flow context essential for understanding vulnerabilities. This oversight leads to two types of misleading outcomes: incorrect conclusions and flawed rationales, collectively undermining the reliability of prior assessments. Therefore, in this paper, we challenge three widely held community beliefs: that LLMs are (i) unreliable, (ii) insensitive to code patches, and (iii) performance-plateaued across model scales. We argue that these beliefs are artifacts of context-deprived evaluations. To address this, we propose CORRECT (Context-Rich Reasoning Evaluation of Code with Trust), a new evaluation framework that systematically incorporates contextual information into LLM-based vulnerability detection. We construct a context-rich dataset of 2,000 vulnerable-patched program pairs spanning 99 CWEs and evaluate 13 LLMs across four model families. Our framework elicits both binary predictions and natural-language rationales, which are further validated using LLM-as-a-judge techniques. Our findings overturn existing misconceptions. When provided with sufficient context, SOTA LLMs achieve significantly improved performance (e.g., 0.7 F1-score on key CWEs), with 0.8 precision. We show that most false positives stem from reasoning errors rather than misclassification, and that while model and test-time scaling improve performance, they introduce diminishing returns and trade-offs in recall. Finally, we uncover new flaws in current LLM-based detection systems, such as limited generalization and overthinking biases.