There has been some theoretical discussion of exotic matter that has negative energy density. It's fun to play around with the possibility and construct (in imagination) things like wormholes that you could walk through to transport instantly to another place and time. However, no evidence of exotic matter actually existing has ever been found.
Gravity is the weakest force in the universe, so it is very difficult to measure directly. The only reason we feel gravity at all is that the other forces tend to cancel out at large distances (compared to the size of an atom or molecule). Small changes in gravitational force are even harder to measure.
Like Cold Fusion, the supposed gravity shield experiments depend on measurement of very small quantities in the presence of a lot of experimental error. It's easy to make a mistake and see something that isn't there because you want to see it. There are lots of examples in the history of science: N-Rays, Canals on Mars, Cold Fusion, etc. (Dare I add the Face on Mars?)
I once had a proud inventor send me a motor design that he claimed put out 100 watts of mechanical power for 99 watts of input electrical power, giving you 1 watt of power for free. It turned out he had a very efficient design so that the mechanical power output was almost (but not quite) equal to the electrical power input. His methods for calculating and measuring the respective powers were inaccurate enough so that it looked like the output power was slightly greater than the input power, but the difference was smaller than the errors in his methods.
It may seem like traditional physicists are too quick to dismiss things that don't fit their narrow worldview, but it's actually just a function of the checks and balances at work when scientists verify each other's studies. Occam's Razor applies unless definitively ruled out. As the late Carl Sagan liked to say, "Extraordinary claims require extraordinary proof."