  The advent of gravitational wave astronomy provides new proving grounds for
testing theories of gravity. Recent work has reinvigorated the study of
bimetric theories of gravity and massive gravity theories. One of the most
interesting predictions of these theories, as well as some string theories, is
the subluminal speed of propagating gravitational waves. Multi-messenger
astronomy provides a unique opportunity to put limits on the difference (either
positive or negative) between the propagation speed of electromagnetic and
gravitational waves from these sources. This paper considers limits from
multi-messenger cases across the planned measurable spectrum: first, the limits
from isolated pulsars based on the current best limits from LIGO on
gravitational wave emission, second, the limits from ultra-compact binaries
that will be visible to a low-frequency space-based gravitational wave
observatory like LISA, and third, limits from super massive black hole binaries
using pulsar timing arrays. The required phase comparison between the
electromagnetic signal and the gravitational wave signal is derived and,
assuming a null result in that comparison, the current bounds on emission are
used to place limits on alternative theories that exhibit propagation delays.
Observations of the pulsars in the most sensitive range of LIGO could put an
upper limit on the graviton mass as low as $10^{-38}\frac{eV}{c^{2}}$ and an
upper limit on the fractional difference between the gravitational wave and
electromagnetic wave speeds as low as $10^{-9}$. This paper shows results from
the initial LIGO limit catalog of known pulsars. The bounds are stronger for
binaries. A LISA-like mission bounds $m_{g}<10^{-40}\frac{eV}{c^{2}}$ and
$\delta<10^{-12}$. A PTA source gives even better bounds of
$m_{g}<10^{-45}\frac{eV}{c^{2}}$ and $\delta<10^{-14}$.
