We study the bounds that space-based gravitational-wave detectors could realistically place on the graviton Compton wavelength ${\ensuremath{\lambda}}_{g}=h/({m}_{g}c)$ by observing multiple inspiralling black hole binaries. Observations of individual inspirals will yield mean bounds ${\ensuremath{\lambda}}_{g}\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}{10}^{15}$ km, but the combined bound from observing $\ensuremath{\sim}50$ events in a two-year mission is about ten times better: ${\ensuremath{\lambda}}_{g}\ensuremath{\simeq}3\ifmmode\times\else\texttimes\fi{}{10}^{16}$ km (${m}_{g}\ensuremath{\simeq}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ eV). The bound improves faster than the square root of the number of observed events, because typically a few sources provide constraints as much as three times better than the mean. This result is only mildly dependent on details of black hole formation and detector characteristics. The bound achievable in practice should be an order of magnitude better than this figure, because our calculations ignore the merger/ringdown portion of the waveform.