We investigate the behavior of a model neuron that receives a biophysically realistic noisy postsynaptic current based on uncorrelated spiking activity from a large number of afferents. We show that, with static synapses, such noise can give rise to inverse stochastic resonance (ISR) as a function of the presynaptic firing rate. We compare this to the case with dynamic synapses that feature short-term synaptic plasticity and show that the interval of presynaptic firing rate over which ISR exists can be extended or diminished. We consider both short-term depression and facilitation. Interestingly, we find that a double inverse stochastic resonance (DISR), with two distinct wells centered at different presynaptic firing rates, can appear.