  We consider the random Cayley graphs of a sequence of finite nilpotent groups
of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly
bounded. For some $k=k(n)$ such that $1\ll\log k \ll \log |G|$, we pick a
random set of generators $S=S(n)$ by sampling $k$ elements $Z_1,\ldots,Z_k$
from $G$ uniformly at random with replacement, and set $S:=\{Z_j^{\pm 1}:1 \le
j\le k \}$. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff
with high probability.
  Some of our results apply to a general set of generators. Namely, we show
that there is a constant $c>0$, depending only on the rank and the nilpotency
class of $G$, such that for all symmetric sets of generators $S$ of size at
most $ \frac{c\log |G|}{\log \log |G|}$, the spectral gap and the
$\varepsilon$-mixing time of the simple random walk $X=(X_t)_{t\geq 0}$ on
Cay$(G,S)$ are asymptotically the same as those of the projection of $X$ to the
abelianization of $G$, given by $[G,G]X_t$. In particular, $X$ exhibits cutoff
if and only if its projection does.
