One of the possible explanations of the so-called Cosmological Lithium Problem (CLIP), i.e., the large overestimate of the primordial 7 Li abundance by the standard Big Bang Nucleosynthesis theory (BBN), is related to an incorrect underestimation of the 7 Be destruction rate by neutron-induced reactions. To verify this possibility, the n + 7 Be reactions have been investigated at n_TOF (CERN) in a wide neutron energy range, taking advantage of the new high-flux experimental area (EAR2) and specifically developed experimental setups. The 7 Be(n, α ) 4 He cross section, measured for the first time from thermal to 10 keV neutron energy, was found consistent with previous estimates. A 10% increase of the 7 Be destruction rate was instead determined on the basis of the 7 Be(n, p) 7 Li cross section measured at n_TOF from thermal to 300 keV neutron energy, a value clearly insufficient to provide a solution to the Cosmological Lithium Problem. Combined together, the two measurements finally rule out neutron-induced reactions on 7 Be as a possible explanation of the CLIP.