Deep image classification models trained on vast amounts of web-scraped data are susceptible to data poisoning - a mechanism for backdooring models. A small number of poisoned samples seen during training can severely undermine a model's integrity during inference. Existing work considers an effective defense as one that either (i) restores a model's integrity through repair or (ii) detects an attack. We argue that this approach overlooks a crucial trade-off: Attackers can increase robustness at the expense of detectability (over-poisoning) or decrease detectability at the cost of robustness (under-poisoning). In practice, attacks should remain both undetectable and robust. Detectable but robust attacks draw human attention and rigorous model evaluation or cause the model to be re-trained or discarded. In contrast, attacks that are undetectable but lack robustness can be repaired with minimal impact on model accuracy. Our research points to intrinsic flaws in current attack evaluation methods and raises the bar for all data poisoning attackers who must delicately balance this trade-off to remain robust and undetectable. To demonstrate the existence of more potent defenders, we propose defenses designed to (i) detect or (ii) repair poisoned models using a limited amount of trusted image-label pairs. Our results show that an attacker who needs to be robust and undetectable is substantially less threatening. Our defenses mitigate all tested attacks with a maximum accuracy decline of 2% using only 1% of clean data on CIFAR-10 and 2.5% on ImageNet. We demonstrate the scalability of our defenses by evaluating large vision-language models, such as CLIP. Attackers who can manipulate the model's parameters pose an elevated risk as they can achieve higher robustness at low detectability compared to data poisoning attackers.