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.
Atsushi Kanazawa
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