So far, there is no experimental evidence that tells against the weak equivalence principle. The consequences of any real violation would be deep indeed: general relativity would instantly be falsified because GTR encodes the principle by saying for any path through spacetime, there is a Lorentz local frame (the "momentarily comoving inertial frame (MCIF)") tangent to the path (i.e. with the same velocity as an observer following that path, no matter how much accelerated that observer may be). In the MCIF, special relativity holds, and so a laboratory attached to this frame would be exactly as in the Galileo's Allegory of Salviati's Ship, i.e. its occupants could do no experiment within the laboratory to detect their motion. This clearly could not be true if "gravity" depended on the composition of objects: the ship's cat would feel different gravity from the instruments on the table and common motion of the lot would not be possible.
It should also be noted that the WEP is what makes the geometrical interpretation of GTR so "real": motion in gravitational fields becomes independent of what is moving in them whether it be a bowling ball or a Brussel sprout, so there is "nothing aside from spacetime itself" determining the freefall motion of a test particle. Contrast this with electromagnetism: Maxwell's equations coupled to the Dirac equation for a particle can also be given a geometric description, but the description depends on the composition (electric charge) of the moving bodies or field. So there would seem to be something else aside from bare "spacetime" setting the motion.
However, most physicists expect GTR to fail at some point and that is must be an approximation to a more complete theory of quantum gravity, which will hopefully be built in the future. It is a distinct possibility that the WEP will only be a classical approximation in such a theory. To make this statement plausible, see this answer by user vnb to the Physics SE question "Does quantum mechanics violate the equivalence principle?". Here Paul Davies is cited giving a thought experiment where Galileo's Tower-Of-Pisa experiment is replaced with different quantum particles of identical initial upwards velocity: classically in the light of the WEP, they should all reach exactly the same height, however, quantum particles tunnel beyond that height, and the extent of tunnelling depends on the "mass" term in the wave equation for the particle in question.
Furthermore, what does then the gravitational mass mean?
If WEP were found defective, then the notion of "gravitational mass" would become highly meaningful and distinct from inertial or rest mass. In this case, "gravitational mass" would be a distinct property analogous to electric charge, it would measure the coupling of an object to the gravitational field. Objects' motions would then depend on their charge to inertial mass ratios as they do in an electromagnetic field. Indeed, one would then tend to think of the gravitational field as "just another field" on top of spacetime instead of the geometry of spacetime itself. Or, more probably, there would be a role for spacetime geometry in setting motions, but it is no longer the only determiner of motions.