  The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring
$(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in
\mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full)
system of parameters, the Frobenius closure $\mathfrak{q}^F$ has
$(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a
suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that
if $R$ is such that the Frobenius closure of the zero submodule in the lower
local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) /
0^F_{H^j_{\mathfrak{m}}(R)}$ is finite length for $0 \le j < \dim(R)$, then
$\operatorname{Fte}(R)<\infty$.
