  We study the interplay between the following types of special non-K\"ahler
Hermitian metrics on compact complex manifolds: it locally conformally
K\"ahler, $k$-Gauduchon, balanced and locally conformally balanced and prove
that a locally conformally K\"ahler compact nilmanifold carrying a balanced or
a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino
and Vezzoni from 2016, this leads to the fact that a compact complex 2-step
nilmanifold endowed with whichever two of the following types of metrics:
balanced, pluriclosed and locally conformally K\"ahler is a torus. Moreover, we
construct a family of compact nilmanifolds in any dimension carrying both
balanced and locally conformally balanced metrics and finally we show a compact
complex nilmanifold does not support a left-invariant locally conformally
hyperK\"ahler structure.
