Counterexample. Consider a $Q$ quadric in $\mathbb CP^3$, let $L_1...,L_n$, $M_1,...,M_n$ be lines on $Q$ so that $L_i\cap L_j=\emptyset$, $M_i\cap M_j=\emptyset$, while $L_i$ intersect $M_j$. Take $n^2$ points $L_i\cap M_j$.
Proof. For a generic projection $\pi: \mathbb P^3\to \mathbb P^2$ both collection of lines $L_i$ and $M_j$ project to (reducible) curves of degree $d$ on $\mathbb P^2$. Their intersection are exactly the projections of the collection of points $L_i\cap M_j$. This gives us a pencil of degree $d$ curves, and in a generic situation a generic curve from the pencil will be smooth.
It is interesting to notice that in the case $n=3$ the above construction is rigid, i.e., it produces a unique example up to projective equivalence of $\mathbb P^3$. Are there some further examples of $9$ points in $\mathbb P^3$ having this property? Are they rigid as well?