  We study almost complex structures on parallelizable manifolds via the rank
of their Nijenhuis tensor. First, we show how the computations of such rank can
be reduced to finding smooth functions on the underlying manifold solving a
system of first order PDEs. On specific manifolds, we find an explicit
solution. Then we compute the Nijenhuis tensor on curves of almost complex
structures, showing that there is no constraint (except for lower
semi-continuity) to the possible jumps of its rank. Finally, we focus on
$6$-nilmanifolds and the associated Lie algebras. We classify which
$6$-dimensional, nilpotent, real Lie algebras admit almost complex structures
whose Nijenhuis tensor has a given rank, deducing the corresponding
classification for left-invariant structures on $6$-nilmanifolds. We also find
a topological upper-bound for the rank of the Nijenhuis tensor for
left-invariant almost complex structures on solvmanifolds of any dimension,
obtained as a quotient of a completely solvable Lie group. Our results are
complemented by a large number of examples.
