In this work, we investigate the connection and compatibility between νn →νn reactions and n-n oscillations based on the SU (3)c×SU (2)L×U (1) symmetry model with additional Higgs triplets. We explore the possibility that the scattering process νn →νn produced by low-energy solar neutrinos gives rise to an unavoidable background in the measurements of n-n oscillations. We focus on two different scenarios, depending on whether the (B − L) symmetry could be broken. We analyze the interplay of the various constraints on the two processes and their observable consequences. In the scenario where both (B + L) and (B − L) could be broken, we point out that if all the requirements, mainly arising from the type-II seesaw mechanism, are satisfied, the parameter space would be severely constrained. In this case, although the masses of the Higgs triplet bosons could be within the reach of a direct detection at the LHC or future high-energy experiments, the predicted n-n oscillation times would be completely beyond the detectable regions of the present experiments. In both scenarios, the present experiments are unable to distinguish a νn →νn reaction event from a n-n oscillation event within the accessible energy range. Nevertheless, if any of the two processes is detected, there could be signal associated with new physics beyond the Standard Model.