The mass and composition of Titan's massive atmosphere, which is dominated by N2 and CH4 at present, have probably varied all along its history owing to a combination of exogenous and endogenous processes. In the present study, we investigate its fate during the Late Heavy Bombardment (LHB) by modeling the competitive loss and supply of volatiles by cometary impacts and their consequences on the atmospheric balance. For surface albedos ranging between 0.1 and 0.7, we examine the emergence of an atmosphere during the LHB as well as the evolution of a primitive atmosphere with various masses and compositions prior to this event, accounting for impact-induced crustal NH3-N2 conversion and subsequent outgassing as well as impact-induced atmospheric erosion. By considering an impactor population characteristic of the LHB, we show that the generation of a N2-rich atmosphere with a mass equivalent to the present-day one requires ammonia mass fraction of 2-5%, depending on surface albedos, in an icy layer of at least 50 km below the surface, implying an undifferentiated interior at the time of LHB. Except for high surface albedos (AS ⩾ 0.7) where most of the released N2 remain frozen at the surface, our calculations indicate that the high-velocity impacts led to a strong atmospheric erosion. For a differentiated Titan with a thin ammonia-enriched crust (⩽5 km) and AS < 0.6 , any atmosphere preexisting before the LHB should be more than 5 times more massive than at present, in order to sustain an atmosphere equivalent to the present-day one. This implies that either a massive atmosphere was formed on Titan during its accretion or that the nitrogen-rich atmosphere was generated after the LHB.