The excitation of the Roper resonance $N^*(1440)$ in $NN$ collisions leading to single and double pion production exhibits a structure in the energy dependence of the total cross section, which does not correspond to the usual opening of the $N^*(1440)$ production channel, but rather to the formation of dibaryon states with $I(J^P) = 0(1^+)$ and $1(0^+)$ at the $N^*(1440)N$ threshold. This is very similar to the situation at the $Delta(1232)N$ threshold, where meanwhile a number of dibaryonic resonances have been found.