Although the Zn2+ cation in Zn center dot Cys(4), Zn center dot Cys(3)His, Zn center dot Cys(2)His(2), and Zn(2)Cys(6) cores of zinc finger (ZO proteins typically plays a structural role, the Zn-bound thiolates in some Zf cores are reactive. Such labile Zf cores can serve as drug targets for retroviral or cancer therapies. Previous studies showed that the reactivity of a Zn-bound thiolate toward electrophiles is significantly reduced if it forms S---NH hydrogen bonds with the backbone amide. However, we found several well-known inactive Zf cores containing Cys ligands with no H-bonding interactions. Here, we show that H bonds from the peptide backbone or bonds from a second Zn cation to Zn-bound S atoms suppress the reactivity not only of these S atoms, but also of Zn-bound S* atoms with no interactions. Indeed, two or more indirect NH---S hydrogen bonds raise the free energy barrier for methylation of a Zn-bound S* in a Cys(4) core more than a direct NH---S* hydrogen bond. These findings help to elucidate why several well-known Zf cores have Cys ligands with no H bonds, but are unreactive. They also help to provide guidelines for distinguishing labile Cys-rich Zn sites from structural ones, which in turn help to identify novel potential Zf drug targets.