In this work, we investigate the potential of Large Language Models (LLMs) for automated short answer scoring. We test zero-shot and few-shot settings, and compare with fine-tuned models and a supervised upper-bound, across three diverse datasets. Our results, in zero-shot and few-shot settings, show that LLMs perform poorly in these settings: LLMs have difficulty with tasks that require complex reasoning or domain-specific knowledge. While the models show promise on general knowledge tasks. The fine-tuned model come close to the supervised results but are still not feasible for application, highlighting potential overfitting issues. Overall, our study highlights the challenges and limitations of LLMs in short answer scoring and indicates that there currently seems to be no basis for applying LLMs for short answer scoring.