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Background: Significant enhancement of sub-barrier fusion cross sections owing to neutron rearrangement with positive Q values were found for many combinations of colliding nuclei. However, several experimental results on fusion reactions were reported recently in which such enhancement has not been observed in spite of a possibility for neutron rearrangement with positive Q values. Purpose: We aim to clarify much better the mechanism of neutron rearrangement in sub-barrier fusion reactions to find the other requirements (beside positive Q values) which favor (or prevent) sub-barrier fusion enhancement. Methods: A channel coupling approach along with the semiclassical model for neutron transfer has been used for analysis of available experimental data on sub-barrier fusion of heavy ions. Results: The role and interplay of different factors determining the enhancement of sub-barrier fusion (such as Q values for neutron rearrangement, properties of collective excitations, and neutron binding energies) have been studied and clarified. Conclusions: (1) Only 1n and 2n transfers with positive Q values have a noticeable impact on sub-barrier fusion. A positive Q value for neutron rearrangement is a necessary but not sufficient requirement for additional sub-barrier fusion enhancement to take place. (2) The "rigidity" of colliding nuclei with respect to collective excitations is important for sub-barrier fusion enhancement due to neutron rearrangement with positive Q values to be clearly visible. (3) The neutron binding energy of the "donor" nucleus has a strong impact only in the case of fusion of light weakly bound nuclei.