A multi-machine study has been carried out to investigate the impact of a strongly bounded wave propagation domain on the Lower Hybrid current drive, a condition which occurs principally in high aspect ratio tokamaks. In this regime, the condition of kinetic resonance can be far above the upper boundary of the propagation domain, and may not be achieved by the usual toroidal upshift. Therefore no tail of fast electrons can be pulled out from the thermal bulk. Nevertheless, while tokamak plasmas are in principle almost transparent to the wave in this regime so-called "unbridgeable spectral gap", full current drive is well achieved for the two tokamaks considered in this study, TRIAM-1M (Zushi et al. Nucl Fusion 43:1600, 2003) and WEST (Bourdelle et al. Nucl Fusion 55:063017, 2015), both characterized by a very large aspect ratio R/a > 5:5. The case of the high aspect ratio tokamak HL-2A (Liu et al. Nucl Fusion 45:S239, 2005) for which the wave propagation domain has also an upper boundary, but close to the resonance condition, is considered by comparison. First principles modeling of the rf-driven current and the fast electron bremsstrahlung using the ALOHA/C3PO/LUKE/R5-X2 chain of codes shows unambiguously that the spectral gap must be already filled at the separatrix in order to reproduce quantitatively observations and some important parametric dependencies. This result is an important milestone in the physics understanding of the Lower Hybrid current drive, highlighting the existence of a powerful and likely universal alternative mechanism to bridge the spectral gap, that is not related to toroidal magnetic refraction. With an initially broad power spectrum, lobes with low parallel refractive indexes that carry most of the plasma current can be absorbed in almost single pass, restoring the full validity of the ray-tracing approximation for describing the propagation of the Lower Hybrid wave in cold plasmas.