Large language models (LLMs) have made rapid improvement on medical benchmarks, but their unreliability remains a persistent challenge for safe real-world uses. To design for the use LLMs as a category, rather than for specific models, requires developing an understanding of shared strengths and weaknesses which appear across models. To address this challenge, we benchmark a range of top LLMs and identify consistent patterns which appear across models. We test 16 well-known LLMs on 874 newly collected questions from Polish medical licensing exams. For each question, we score each model on the top-1 accuracy and the distribution of probabilities assigned. We then compare with factors including question difficulty for humans, question length, and the scores of the other models. LLM accuracies were positively correlated pairwise (0.39 to 0.58). Model performance was also correlated with human performance (0.09 to 0.13), but negatively correlated to the difference between the question-level accuracy of top-scoring and bottom-scoring humans (-0.09 to -0.14). The top output probability and question length were positive and negative predictors of accuracy respectively (p < 0.05). The top scoring LLM, GPT-4o Turbo, scored 84%, with Claude Opus, Gemini 1.5 Pro and Llama 3/3.1 between 74% and 79%. We found evidence of similarities between models in which questions they answer correctly, as well as similarities with human test takers. Larger models typically performed better, but differences in training, architecture, and data were also highly impactful. Model accuracy was positively correlated with confidence, but negatively correlated with question length. We find similar results with older models, and argue that these patterns are likely to persist across future models using similar training methods.