The recently introduced TabPFN pretrains an In-Context Learning (ICL) transformer on synthetic data to perform tabular data classification. In this work, we extend TabPFN to the fine-tuning setting, resulting in a significant performance boost. We also discover that fine-tuning enables ICL-transformers to create complex decision boundaries, a property regular neural networks do not have. Based on this observation, we propose to pretrain ICL-transformers on a new forest dataset generator which creates datasets that are unrealistic, but have complex decision boundaries. TabForest, the ICL-transformer pretrained on this dataset generator, shows better fine-tuning performance when pretrained on more complex datasets. Additionally, TabForest outperforms TabPFN on some real-world datasets when fine-tuning, despite having lower zero-shot performance due to the unrealistic nature of the pretraining datasets. By combining both dataset generators, we create TabForestPFN, an ICL-transformer that achieves excellent fine-tuning performance and good zero-shot performance.