It's not straightforward to test conservation of momentum experimentally, and many experiments that seem like tests really aren't. For example, in a Newtonian system of identical particles that interact through collisions, conservation of momentum follows simply from Galilean invariance and the symmetry of the collisions in the center of mass frame. For reasons such as these, many freshman-physics tests of conservation of momentum may not actually test it, even approximately -- that is, they may not present even the logical possibility of falsifying the conservation law.
An added problem is that both of the fundamental theories of physics, quantum mechanics and general relativity, have local conservation of the energy-momentum four-vector pretty thoroughly embedded in their structures. In general, it's hard to test a hypothesis unless you have a test theory that is consistent with the failure of the hypothesis.
In the case of GR, we have the PPN formalism, which, although not really a scientific theory, does allow for nonconservation of momentum. The best test I know of within this framework is a lunar laser ranging experiment by Bartlett (1986), which verified the equality of active and passive gravitational mass to a precision of about $10^{-10}$. The validity of this test depended on the inhomogeneity of the moon -- otherwise, for reasons similar to those described in the first paragraph, an anomalous acceleration is forbidden by symmetry. More recent observations of pulsars constrain the momentum-nonconserving PPN parameter $|\alpha_3|$ to be less than $5\times10^{-16}$ (Bell 1995).
What about tests at the microscopic scale, in the electromagnetic sector? Of course it's hard to imagine theoretically how conservation of momentum could fail, since it seems to follow directly from translation invariance and Noether's theorem. But this isn't the same thing as verifying it experimentally.
Have there been nongravitational tests at the macroscopic scale, e.g., empirical limits on spontaneous self-accelerations of inhomogeneous kilogram-scale masses? (Seems like the kind of thing the Eot-Wash group would do.)
The interpretation of this kind of thing depends on whether or not your test theory allows Lorentz violation. E.g., the PPN formalism explicitly allows for both momentum nonconservation and Lorentz violation. If Lorentz invariance is valid, then any test of conservation of energy is also a test of conservation of momentum. So there might be one bound on nonconservation of momentum if you don't assume Lorentz invariance, and some other, tighter bound if you do.
[EDIT] It seems like the atomic-physics tests are conventionally described as tests of local position invariance (LPI), although by Noether's theorem that's equivalent to conservation of energy-momentum. The highest-precision experiments compare the rates of atomic clocks of different atomic species and watch for a variation in the ratio of their rates over time. One can also test for universality of gravitational redshifts, or compare atomic clocks (microscopic) with electromagnetic resonators (macroscopic). Some recent papers are Guéna 2012 and Agachev 2010. When I asked the question, I hadn't hit upon the right search term to turn up these experiments. I'd still be interested in seeing a synoptic answer, or one that touched on the strong-force sector, or one that provided an interesting review of the history of such tests.
Agachev 2010, http://link.springer.com/article/10.1134%2FS0202289311010026#page-1
Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21, summarized in Will, http://relativity.livingreviews.org/Articles/lrr-2006-3/
Bauch, http://ebookbrowse.com/2002-bauch-weyers-phys-rev-d65-pdf-d370051021
Bell, "A Tighter Constraint on post-Newtonian Gravity using Millisecond Pulsars," http://arxiv.org/abs/astro-ph/9507086
Guéna 2012, http://arxiv.org/abs/1205.4235
