  In this paper we show that a nodal complete intersection threefold $X$ in
$\mathbb{P}^{3+c}$ with defect, but without induced defect, has at least
$\sum_{i\leq j} (d_i-1)(d_j-1)$ nodes, provided either $c=2$ or
$d_c>\sum_{i=1}^{c-1} d_i$ holds.
