No, that's not true. Regardless your local network speeds, you are not going faster than your provider allows. While FiOS is certainly faster than cable or DSL, as seen in this
Verizon FiOS Internet comparison, the best download speed you can get (and certainly these are ideal "marketing figures" for their most expensive package) is 125Mb and the best upload is 500Mb - that's 87.5% and 50%
slower (AKA, a bottleneck) than a 1Gb network.
So yeah, if you have a 1Gb local network, you MIGHT see some improvement over the typical 100Mb network for downloads. And let's be realistic here, fast upload speeds only affect a few users out there.
The technology is there - fiber has the capability, but the ISPs have yet to tap its potentials.