  It is well known that cohomology of any non-trivial 1-dimensional local
system on a nilmanifold vanishes (this result is due to L. Alaniya). A complex
nilmanifold is a quotient of a nilpotent Lie group equipped with a
left-invariant complex structure by an action of a discrete, co-compact
subgroup. We prove a Dolbeault version of Alaniya's theorem, showing that the
Dolbeault cohomology of a nilpotent Lie algebra with coefficients in any
non-trivial 1-dimensional local system vanishes. Note that the Dolbeault
cohomology of the corresponding local system on the manifold is not necessarily
zero. This implies that the twisted version of Console-Fino theorem is false
(Console-Fino proved that the Dolbeault cohomology of a complex nilmanifold is
equal to the Dolbeault cohomology of its Lie algebra). As an application, we
give a new proof of a theorem due to H. Sawai, who obtained an explicit
description of LCK nilmanifolds. An LCK structure on a manifold $M$ is a
K\"ahler structure on its cover $\tilde M$ such that the deck transform map
acts on $\tilde M$ by homotheties. We show that any complex nilmanifold
admitting an LCK structure is Vaisman, and is obtained as a compact quotient of
the product of a Heisenberg group and the real line.
