While the broad-lined Type Ic supernovae (SN Ic-bl) associated with long-duration gamma-ray bursts (GRBs) have been studied, we do not fully understand the conditions that lead to each kind of explosion in a massive star. Here we show clues as to the production mechanism of GRBs by comparing the chemical abundances at the sites of 5 nearby (z < 0.25) broad-lined SN Ic that accompany nearby GRBs with those of 12 nearby (z < 0.14) broad-lined SN Ic that have no observed GRBs. We show that the oxygen abundances at the GRB sites are systematically lower than those found near ordinary broad-lined SN Ic. A unique feature of this analysis is that we present new spectra of the host galaxies and analyze the measurements of both samples in the same set of ways, using 3 independent metallicity diagnostics. We demonstrate that neither SN selection effects (SN found via targeted vs. non-targeted surveys) nor the choice of strong-line metallicity diagnostic can cause the observed trend. Though our sample size is small, the observations are consistent with the hypothesis that low metal abundance is the cause of some massive stars becoming SN-GRB. We derive a cut-off metallicity of 0.2−0.6 Z⊙, with the exact value depending on the adopted metallicity scale and solar abundance value.