  We describe, for various degenerations $S\to \Delta$ of quartic $K3$ surfaces
over the complex unit disk (e.g., to the union of four general planes, and to a
general Kummer surface), the limits as $t\in \Delta^*$ tends to 0 of the Severi
varieties $V_\delta(S_t)$, parametrizing irreducible $\delta$-nodal plane
sections of $S_t$. We give applications of this to (i) the counting of plane
nodal curves through base points in special position, (ii) the irreducibility
of Severi varieties of a general quartic surface, and (iii) the monodromy of
the universal family of rational curves on quartic $K3$ surfaces.
