This paper investigated the finite-time stability (FTS) and nearly fixed-time stability (nFxTS) of nonlinear impulsive systems with destabilizing impulses. The concept of nFxTS is different from FxTS, as the former may be just asymptotically stable but the latter must be FTS. A Lyapunov inequality with linear terms has been used to derive sufficient conditions based on the dwell time (DT) and average dwell time (ADT) properties of impulsive sequences to ensure FTS/FxTS of the system. Theorems are constructed based on the initial state of the system, which belongs to either-neighbourhood(nbh) or outside-nbh of the origin. When the system starts within −nbh of the origin then FTS can be achieved by the system following conditions on DT and ADT of impulsive sequences. The same is true for nFxTS of the system when it starts outside-nbh of the origin. We investigated the condition on DT for which nFxTS is not possible, but FTS is possible only if > 0 is bounded. The main results of this paper are applied to the synchronization problem of impulsive neural networks with destablizing impulses. Finally, a numerical example of an impulsive neural network is given to illustrate the validity of the proposed theoretical results.