Huang, Klemm and Quackenbush computed the BPS invariants of quintic 3-fold for some low genera and degrees via the BCOV technique in http://arxiv.org/abs/hep-th/0612125. One can get the GW invariants from their data.

I think the bound is not a theorem, but an observation. We often assume such a vanishing condition to solve the BCOV holomorphic anomaly equation.

Huang, Klemm and Quackenbush computed the BPS invariants of the quintic 3-fold for low genera via the BCOV technique in http://arxiv.org/abs/hep-th/0612125. We can easily convert their data to get the GW invariants.

I think the bound is not a theorem, but an observation. We often assume such a vanishing condition to effectively solve the BCOV holomorphic anomaly equations.