Annihilation of extremely energetic cosmic neutrinos on the relic-neutrino background can give rise to absorption lines at energies corresponding to formation of the electroweak gauge boson Z(0). The positions of the absorption dips are set by the masses of the relic neutrinos. Suitably intense sources of extremely energetic (10(21)-10(25)-eV) cosmic neutrinos might therefore enable the determination of the absolute neutrino masses and the flavor composition of the mass eigenstates. Several factors-other than neutrino mass and composition-distort the absorption lines, however. We analyze the influence of the time evolution of the relic-neutrino density and the consequences of neutrino decay. We consider the sensitivity of the line shape to the age and character of extremely energetic neutrino sources, and to the thermal history of the Universe, reflected in the expansion rate. We take into account Fermi motion arising from the thermal distribution of the relic-neutrino gas. We also note the implications of Dirac vs. Majorana relics, and briefly consider unconventional neutrino histories. We ask what kinds of external information would enhance the potential of cosmic-neutrino absorption spectroscopy, and estimate the sensitivity required to make the technique a reality.