Knowledge graph (KG) foundation models (KGFMs) are zero-shot generalizers: trained once, they can predict links on unseen graphs without retraining. However, understanding when and how they can robustly generalize across KGs is still an open question. In this paper, we shed some light on their generalization mechanisms highlighting how their performance on unseen KGs is not uniform when it comes to partially seen links, which we call *half-links*. In fact, we show that to predict a test triple (h,r,t) it might suffice in practice to have observed the half-link (h,r) or (r,t) in the inference graph. This yields a taxonomy of four scenarios when combinations of these half-links are observed or not. In a rigorous stratified analysis over these scenarios, we reveal that SoTA KGFMs use seen half-links for predictions, while unseen half-links pose different challenges. As such, our finer-grained taxonomy can be a diagnostic protocol for robust KGFM generalization and highlights where novel KGFMs can improve.