Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to do so. Focusing on the situation in which the unstimulated neuron is quiescent but close to a switching point for oscillations, we numerically and dynamically analyze STR treating each noise-driven spike occurrence as a time localized event in a model neuron. We study both the averaged properties as well as individual features of spike-evoking epochs. The effects of interactions between spikes is minimized by selecting signals that generate spikes with relatively long interspike intervals. We study two distinct cases characterized by either a smooth or discontinuous dependence of frequency on the applied current. In the setting of noise-induced spiking the frequency content of the input signal has little impact on STR. Rather, increased STR is observed for a certain increase in the average amount of current delivered during a fixed time interval in combination with a favorable time profile. These computational results are complemented by an analytical approximation for the density of the phase difference of two coupled stochastically forced oscillators, considering different relative sizes of the intrinsic and extrinsic noises. When common noise forcing dominates, there is a stronger probability of synchronization related to STR. In contrast, if the intrinsic noises are not of identical strength, there is a synchronized lag between oscillations.