Fine-tuning pretrained models is a standard and effective workflow in modern machine learning. However, robust fine-tuning (RFT), which aims to simultaneously achieve adaptation to a downstream task and robustness to adversarial examples, remains challenging. Despite the abundance of non-robust pretrained models in open-source repositories, their potential for RFT is less understood. We address this knowledge gap by systematically examining RFT from such non-robust models. Our experiments reveal that fine-tuning non-robust models with a robust objective, even under small perturbations, can lead to poor performance, a phenomenon that we dub \emph{suboptimal transfer}. In challenging scenarios (eg, difficult tasks, high perturbation), the resulting performance can be so low that it may be considered a transfer failure. We find that fine-tuning using a robust objective impedes task adaptation at the beginning of training and eventually prevents optimal transfer. However, we propose a novel heuristic, \emph{Epsilon-Scheduling}, a schedule over perturbation strength used during training that promotes optimal transfer. Additionally, we introduce \emph{expected robustness}, a metric that captures performance across a range of perturbations, providing a more comprehensive evaluation of the accuracy-robustness trade-off for diverse models at test time. Extensive experiments on a wide range of configurations (six pretrained models and five datasets) show that \emph{Epsilon-Scheduling} successfully prevents \emph{suboptimal transfer} and consistently improves expected robustness.