Deep image classification models trained on large amounts of web-scraped data are vulnerable to data poisoning, a mechanism for backdooring models. Even a few poisoned samples seen during training can entirely undermine the model's integrity during inference. While it is known that poisoning more samples enhances an attack's effectiveness and robustness, it is unknown whether poisoning too many samples weakens an attack by making it more detectable. We observe a fundamental detectability/robustness trade-off in data poisoning attacks: Poisoning too few samples renders an attack ineffective and not robust, but poisoning too many samples makes it detectable. This raises the bar for data poisoning attackers who have to balance this trade-off to remain robust and undetectable. Our work proposes two defenses designed to (i) detect and (ii) repair poisoned models as a post-processing step after training using a limited amount of trusted image-label pairs. We show that our defenses mitigate all surveyed attacks and outperform existing defenses using less trusted data to repair a model. Our defense scales to joint vision-language models, such as CLIP, and interestingly, we find that attacks on larger models are more easily detectable but also more robust than those on smaller models. Lastly, we propose two adaptive attacks demonstrating that while our work raises the bar for data poisoning attacks, it cannot mitigate all forms of backdooring.