Barker and Ortiz found unusual falloff effects in the flux coefficients of the competitive unimolecular reactions of 2-methylhexyl radicals, and they concluded that this might have important effects on the rate constants of reactions with higher thresholds. To study this effect, we carried out master equation calculations of the same reaction system to learn whether this effect shows up in measurable rate constants, and the answer is yes. We also studied specially designed mechanisms to reveal that the various reactive pathways connecting the reagents can have a large effect on the rate constants, causing them to be quite different than if the reactions proceeded independently, and that reactions with significantly higher barriers may nevertheless have larger rate constants. This provides a new perspective for interpreting and predicting the kinetics of competitive unimolecular reactions.