The Sn118,124(p,t)Sn116,122 reactions have been investigated in high-resolution experiments at incident proton energies of 24.6 and 25 MeV, respectively. Angular distributions for 55 transitions to levels of Sn116 and 63 transitions to levels of Sn122, up to excitation energies of ~3.850 and ~4.000 MeV, respectively, have been measured. The spin and parity identification was carried out by means of a distorted-wave Born approximation (DWBA) analysis, performed by using conventional Woods-Saxon potentials. A shell-model study of Sn116 and Sn122 nuclei was performed using a realistic two-body effective interaction derived from the CD-Bonn nucleon-nucleon potential. The doubly magic nucleus Sn132 was assumed as a closed core, with the 16 and 10 valence neutron holes occupying the five levels of the 50-82 shell. The energy spectra have been calculated and compared with the experimental ones, and the theoretical two-nucleon spectroscopic amplitudes, evaluated in a truncated seniority space, have been used in the microscopic DWBA calculation of some cross-section angular distributions of both reactions.