The effects of tunneling between two parallel two-dimensional electron gases in n-InGaAs/GaAs nanostructures with strongly coupled double quantum wells with a change in the in-plane component of a tilted magnetic field (up to B ‖ = 9.0 T) in the temperature range T = 1.8–70.0 K are investigated. A nonmonotonic temperature dependence of the inverse quantum lifetime τ q − (T) is obtained from analysis of the dependence of the longitudinal resistance on the parallel component of the tilted magnetic field at fixed temperatures, ρ xx (B ‖, T). The quadratic portion of this dependence is found to be due to the contribution of inelastic electron-electron scattering. The decrease in the inverse quantum lifetime τ q − (T) at T > 0.1T F cannot be described within known theories; it seems, it is not related to the processes of electron momentum relaxation.