Given a curve $C \subset \mathbb{P}^3$ does there exists an IRREDUCIBLE family of curves in $\mathbb{P}^3$ such that $C$ is a general fiber and there exists a fiber which is the union of lines (not necessarily reduced)? In other words does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)?