When you say multiple simultaneous action potentials I assume that the stimuli for all of them are temporally overlapping. In such case a neuron can integrate the different stimuli and launch an action potential. Of the multiple stimuli some can be excitatory while others can be inhibitory. The net response would be an integration of all the signals [ref].
However, if there is a gap in the stimuli then the cell may not react as it would be in its refractory period (which results until the ions levels are back to their original state and the membrane potential is restored).
There can be long delays, which depends on various factors such as ion deficiency, ATP deficiency (ATP is needed by Na+/K+ ATPase) or neuropathological conditions. In case of nervous signaling over many neurons then synapses can suffer from synaptic fatigue, which results because of depletion of neurotransmitters.
Maximum AP propagation speed/Nerve conduction velocity = ~120m/s (Can't find original source. From wikipedia)
Longest myelinated axon (sciatic nerve in humans) = ~ 1m
Time required to reach synapse = ~8ms
Time required for repolarization = ~5µs
So multiple action potentials (~160), are possible (lets call it situation-X).
The squid giant axon is ~500µm. I believe it is also non-myelinated. In this case if NCV is less than 100m/s then situation-X can happen. NCV can go as low as 0.5m/s (again the reference is wikipedia as I cannot find the original source).
Having said that, situation-X can also cause synaptic fatigue if the firing rate (AP frequency) goes beyond a certain value. This value would depend on how much neurotransmitter is released per AP, how much is in storage and the uptake rate.