In 1987 R. Kobayashi proposed a conjectural mechanism for Ricci-flat metrics on $\K 3$ surfaces to degenerate into unions of Tian-Yau and Taub-NUT spaces. See Cases (i) and (ii) on p.223 in [Kob90]. Our main result in this paper verifies Kobayashi’s expectation at the level of hyperkähler structures. However, at least in certain cases, Kobayashi also proposed an identification of these limits with type II polarized degenerations of complex structures on $\K 3$ surfaces.