  In this paper we deal with the following question: is it true that any
bounded smooth pseudoconvex domain in $\mathbb{C}^n$ whose boundary contains a
$q$-dimensional complex manifold $M$ necessarily has a noncompact
$\overline\partial$-Neumann operator $N_q$ ($1\leq q\leq n-1$)?
  We prove that a smooth bounded pseudoconvex domain
$\Omega\subseteq\mathbb{C}^3$ with a one-dimensional complex manifold $M$ in
its boundary has a noncompact Neumann operator on $(0,1)$-forms, under the
additional assumption that $b\Omega$ has finite regular D'Angelo $2$-type at a
point of $M$, improving previous results of Fu, \c{S}ahuto\u{g}lu, and Straube.
