  We construct the quantum mechanical model of the COW experiment assuming that
the underlying space time has a granular structure, described by a canonical
noncommutative algebra of coordinates $x^{\mu}$. The time-space sector of the
algebra is shown to add a mass-dependent contribution to the gravitational
acceleration felt by neutron deBrogli waves measured in a COW experiment. This
makes time-space noncommutativity a potential candidate for an apparent
violation of WEP even if the ratio of the inertial mass $m_{i}$ and
gravitational mass $m_{g}$ is a universal constant. The latest experimental
result based on COW principle is shown to place an upper-bound several orders
of magnitude stronger than the existing one on the time-space noncommutative
parameter. We argue that the evidence of NC structure of space-time may be
found if the COW-type experiment can be repeated with several particle species.
