Phase shifts and inelasticity parameters for $NN$ scattering in the partial-wave channels ${}^3S_1$--${}^3D_1$ and ${}^1S_0$ at energies $T_{rm lab}$ from zero to about 1 GeV are described within a unified $NN$ potential model assuming the formation of isoscalar and isovector dibaryon resonances near the $NN^*(1440)$ threshold. Evidence for these near-threshold resonances is actually found in the recent WASA experiments on single- and double-pion production in $NN$ collisions. There, the excitation of the Roper resonance $N^*(1440)$ exhibits a structure in the energy dependence of the total cross section, which corresponds to the formation of dibaryon states with $I(J^pi)=0(1^+)$ and $1(0^+)$ at the $NN^*(1440)$ threshold. These two $S$-wave dibaryon resonances may provide a new insight into the nature of the strong $NN$ interaction at low and intermediate energies.