If point masses could be generated in a controlled manner distorting the fabric of space-time objects could be made to move in any direction relative to each other and not simply attract as is normally experienced. Imagine a rubber sheet stretched tight with a heavy mass [m1] at its center creating a gravitational well. Now add a point mass some distance away [m2] that distorts the rubber sheet creating a second well. m2 and objects close to it would move toward it and away from the m1. Could this be the basis for an anti-gravity effect? Envision water flowing across the surface of a rubber sheet distorted by a mass at its center; now place another mass [a marble] on the sheet some distance from the center causing another distortion, the flow of water close to the second mass changes, and flows away from the center and toward the second mass. If this distortion is made in space the effect would be anti-gravity would it not?
An example of the sort of system you describe would be the Earth and the Moon, with the Earth playing the part of your mass $m_1$ and the Moon $m_2$. Neither of these are point masses, but courtesy of Gauss' law we know that the gravitational field of a sphere is the same as the gravitational field of a point mass provided you are farther away than the radius of the sphere.
When the Moon is overhead it does indeed slightly reduce the force we feel from the Earth's gravity, but this is simply because the Earth pulls us one way and the Moon pulls us in the opposite direction. No physicist would describe this as anti-gravity.
Given that you mention water it's worth mentioning that the obvious effect of the Moon's gravity is the tides. If you progressively made the Moon more massive and brought it closer the tides would get bigger, and there would come a point where the Moon would pull water off the Earth (though by time it would be pulling the Earth to bits as well!).