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We study the deformations of a smooth curve $C$ on a smooth projectivethreefold $V$, assuming the presence of a smooth surface $S$ satisfying $C\subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a newsufficient condition for a first order infinitesimal deformation of $C$ in $V$to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, wegive a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of$(-2)$-curves and elliptic curves on $S$. Applying this result, we prove thatthe Hilbert scheme $\operatorname{Hilb}^{sc} V_4$ of smooth connected curves ona smooth quartic threefold $V_4$ contains infinitely many genericallynon-reduced irreducible components, which are variations of Mumford's examplefor $\operatorname{Hilb}^{sc} \mathbb P^3$.

Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface