Being motivated by the observation of an anomaly in the polarization of protons in $\ensuremath{\gamma}d\ensuremath{\rightarrow}\mathrm{pn}$, we have investigated the possible existence of deeply bound dibaryon states. The nonrelativistic one-boson-exchange potential model ($\ensuremath{\pi}$, $\ensuremath{\eta}$, $\ensuremath{\rho}$, and $\ensuremath{\omega}$ exchanges) for two $\ensuremath{\Delta}$ isobars gives binding energies around 100 MeV for $I=0$, $J=3$, and $I=3$, $J=0$. Existence of this $I=0$ dibaryon resonance reproduces experimentally observed cross sections in the $I=0$, $\mathrm{NN}\ensuremath{\rightarrow}d\ensuremath{\pi}\ensuremath{\pi}$ and $\mathrm{NN}\ensuremath{\rightarrow}\mathrm{NN}\ensuremath{\pi}\ensuremath{\pi}$ channels.