It has been argued that Uranus and Neptune could reach their present sizes only at the expense of an initial amount of mass in their accretion zones far exceeding their current masses [e.g., V.S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets (translated from Russian (1972) by the Israel Program for Scientific Translation, Jerusalem), 1969, and in The Motion, Evolution of Orbits, and Origin of Comets (G. A. Chebotarev, E. I. Kazimirchak-Polonskaya, and B. G. Marsden, Eds.), pp. 329-334, IAU Symposium No. 45, 1972]. From a numerical analysis, it is determined that such a scenario would have led to a heavy bombardment of the inner planetary region by cometary bodies as well as the formation of a cometary reservoir (the so-called "Oort cloud") at very large heliocentric distances. The cometary influx event can be characterized by the arrival of two different populations of stray bodies. At the beginning, the stray population is made up of "outer planetary region" comets, transferred ffrom the Uranus-Neptune region to the region of the terrestrial planets by gravitational perturbations of the Jovian planets. These comets are characterized for having low-inclination orbits and, once they reach the region of the terrestrial planets, for being mainly under the gravitational influence of Jupiter. Later, "Oort-cloud" comets, driven into the inner planetary region by stellar perturbations, become the dominant component among the incoming comets. The crossover from a outer planetary region comets-dominated regime to a Oort cloud comets-dominated one is found to occur at t∼1.5-2AE. Besides the Oort cloud population, a residual population of mass about 10-5 times the initial mass is found to remain in the outer planetary region at present. Most of the survivors move on orbits with perihelia in the Uranus-Neptune region and aphelia beyond Neptune's orbit, reaching heliocentric distances up to hundreds or even thousands AU. We thus propose two possible cometary sources: (1) A cometary reservoir-the "Oort cloud"-of rather spherical structure with a concentration of aphelion points at several times 104 AU. Members of the cloud are subjected to the perturbing influence of passing stars. (2) A rather flat system of cometary bodies-the "cometary belt" - subjected to planetary perturbations but where stellar perturbations have a negligible role due to the larger binding energies of belt comets. Despite their long dynamical time scales, they will be finally ejected or scattered to the region of the terrestrial planets. In the latter case, such belt objects will appear as short-period comets. Therefore, a belt of residual bodies in the outer fringes of the planetary region may be a likely source of the observed family of short-period comets.