If two good solvents become poor for a polymer when mixed, the solvent pair is called a cononsolvent pair. The sharp reentrant coil-to-globule-to-coil transition of a $\mathrm{\text{poly}}(N\mathrm{\text{-isopropylacrylamide}})$ chain observed in the mixed solvent of water and methanol is shown to be caused by the competitive hydrogen bonding by water and methanol molecules onto the polymer chain. On the basis of a new statistical-mechanical model for competitive hydrogen bonds, the mean square end-to-end distance is theoretically calculated and compared with experiment. The chain sharply collapses at the molar fraction $x_m\simeq 0.2$ of methanol, stays collapsed up to $x_m\simeq 0.4$, and finally recovers the swollen state at $x_m\simeq 0.6$. Such a reentrant coil-globule transition takes place because the total number of hydrogen bonds along the chain exhibits a similar square-well-type depression as a result of the competition.