Large Language Models (LLMs) recently achieved great success in medical text summarization by simply using in-context learning. However, these recent efforts do not perform fine-grained evaluations under difficult settings where LLMs might fail. They typically report performance scores over the entire dataset. Through our benchmarking study, we show that LLMs show a significant performance drop for data points with high concentration of out-of-vocabulary (OOV) words or with high novelty. Vocabulary adaptation is an intuitive solution to this vocabulary mismatch issue where the LLM vocabulary gets updated with certain expert domain (here, medical) words or subwords. An interesting finding from our study is that Llama-3.1, even with a vocabulary size of around 128K tokens, still faces _over-fragmentation_ issue with medical words. To that end, we show vocabulary adaptation helps improve the LLM summarization performance even in difficult settings. Through extensive experimentation of multiple vocabulary adaptation strategies, two continual pretraining strategies, and three benchmark medical summarization datasets, we gain valuable insights into the role of vocabulary adaptation strategies for customizing LLMs to the medical domain. We also performed a human evaluation study with medical experts where they found that vocabulary adaptation results in more relevant and faithful summaries. Our codebase is made publicly available at https://github.com/gb-kgp/LLM-MedicalSummarization-Benchmark.