  Let $G$ be a linear algebraic group acting linearly on a vector space $V$,
and let $k[V]^G$ be the corresponding algebra of invariant polynomial
functions. A separating set $S \subseteq k[V]^G$ is a set of polynomials with
the property that for all $v,w \in V$, if there exists $f \in k[V]^G$
separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$.
  In this article we consider the action of $G = \mathrm{SL}_2 \times
\mathrm{SL}_2$ on the $\mathbb{C}$-vector space $M_{2,2}^n$ of $n$-tuples of $2
\times 2$ matrices by multiplication on the left and the right. Minimal
generating sets $S_n$ of $\mathbb{C}[M_{2,2}^n]^G$ are known, and $|S_n| =
\frac{1}{24}(n^4-6n^3+23n^2+6n)$. In recent work, Domokos showed that $S_n$ is
a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a
separating set. Our main result shows that any separating set for
$\mathbb{C}[M_{2,2}^n]^G$ has cardinality $\geq 5n-9$. In particular, there is
no separating set of size $\dim(\mathbb{C}[M_2^n]^G) = 4n-6$ for $n \geq 4$. We
also consider the action of $G= \mathrm{SL}_l(\mathbb{C})$ on $M_{l,n}$ by left
multiplication. In that case the algebra of invariants has a minimum generating
set of size $\binom{n}{l}$ and dimension $ln-l^2+1$. We show that a separating
set for $\mathbb{C}[M_{l,n}]^G$ must have size at least $(2l-2)n-2(l^2-l)$. In
particular, $\mathbb{C}[M_{l,n}]^G$ does not contain a separating set of size
$\dim(\mathbb{C}[M_{l,n}]^G)$ for $l \geq 3$ and $n \geq l+2$. We include an
interpretation of our results in terms of representations of quivers, and make
a conjecture generalising the Skowronski-Weyman theorem.
