One can imagine that each action potential causes a small amount of $\ce{Na+}$ goes inside the cell, and a small amount $\ce{K+}$ goes outside the cell, thus weakening the electrochemical gradient of both ions. If each action potential has (approximately) the same flux of $\ce{Na+}$ and $\ce{K+}$ then higher frequency of action potentials means more flux, thus a quicker depletion of the electrochemical gradients. The absolute refractory means there is a maximum firing frequency.
Thus the Sodium-Potassium pump only needs to be able to "recharge" the maximal possible depletion of ion's gradients. If there was no absolute refractory period then theoretically the action potential frequency could be faster than the Sodium-Potassium pump can keep up. A abnormally fast firing rate in theory could deplete the potassium gradient and sodium gradient thus would result in a fatigued neuron.
Edited to add in reference to the conjecture:
Absolute refractory periods are cause by sodium gate inactivation, thus no matter how much current one adds the sodium gates will not open until sodium inactivation ends. However, over long periods of time (in seconds) and in the absences of a ionic pump, the lack of sodium (and potassium) gradients will cause fatigue.
However, "fatigued" neurons may not act like one expects.
