  In the present paper, we study the notion of the Schur multiplier
$\mathcal{M}(L)$ of an $n$-Lie superalgebra $L$, and prove that $\dim
\mathcal{M}(L) \leq \sum_{i=0}^{n} {m\choose{i}} \mathcal{L}(n-i,k)$, where
$\dim L=(m|k)$, $\mathcal{L}(0,k)=1$ and $\mathcal{L}(t,k) =
\sum_{j=1}^{t}{{t-1}\choose{j-1}} {k\choose j}$, for $1\leq t\leq n$. Moreover,
we obtain an upper bound for the dimension of $\mathcal{M}(L)$ in which $L$ is
a nilpotent $n$-Lie superalgebra with one-dimensional derived superalgebra. It
is also provided several inequalities on $\dim\mathcal{M}(L)$ as well as an
$n$-Lie superalgebra analogue of the converse of Schur's theorem.
