The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a
positive characteristic invariant obtained by comparing the powers of $\a$ with
the Frobenius powers of $J$. We show that under mild assumptions, we can detect
the containment in the integral closure or the tight closure of a parameter
ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms
of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are
zero-dimensional ideals, and $J$ is generated by a system of parameters. We
prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and
also when $\a$ and $J$ are generated by homogeneous systems of parameters in a
Cohen-Macaulay graded $k$-algebra.