Code adaptation is a fundamental but challenging task in software development, requiring developers to modify existing code for new contexts. A key challenge is to resolve Context Adaptation Bugs (CtxBugs), which occurs when code correct in its original context violates constraints in the target environment. Unlike isolated bugs, CtxBugs cannot be resolved through local fixes and require cross-context reasoning to identify semantic mismatches. Overlooking them may lead to critical failures in adaptation. Although Large Language Models (LLMs) show great potential in automating code-related tasks, their ability to resolve CtxBugs remains a significant and unexplored obstacle to their practical use in code adaptation. To bridge this gap, we propose CtxBugGen, a novel framework for generating CtxBugs to evaluate LLMs. Its core idea is to leverage LLMs' tendency to generate plausible but context-free code when contextual constraints are absent. The framework generates CtxBugs through a four-step process to ensure their relevance and validity: (1) Adaptation Task Selection, (2) Task-specific Perturbation,(3) LLM-based Variant Generation and (4) CtxBugs Identification. Based on the benchmark constructed by CtxBugGen, we conduct an empirical study with four state-of-the-art LLMs. Our results reveal their unsatisfactory performance in CtxBug resolution. The best performing LLM, Kimi-K2, achieves 55.93% on Pass@1 and resolves just 52.47% of CtxBugs. The presence of CtxBugs degrades LLMs' adaptation performance by up to 30%. Failure analysis indicates that LLMs often overlook CtxBugs and replicate them in their outputs. Our study highlights a critical weakness in LLMs' cross-context reasoning and emphasize the need for new methods to enhance their context awareness for reliable code adaptation.