I agree with the idea that empathy, in any given person, exists as a limited commodity.
I believe that a person is by nature a limited being; there are bounds to all dimensions of experience, feeling, etc. Once a person gets into the appearance of having unlimited any-quality-within-the-self, it seems to me like we are now talking about gods or fictions. Or narcissists, I suppose.
I have read that our daily ability to self-discipline is limited; that it shows a decline in performance over time when taxed. (Yes, I've also read articles that decry those same studies; seems like the jury is out on this topic. However, I've never seen a study demonstrating a human's unlimited quantity of -- well, anything.) I know for a fact that attributes of mine, such as patience or anger, certainly feel and appear limited as I engage in or with them. My general level of intelligence is limited by my brain and genes. My stamina and physical endurance are limited, as I begin to feel exhausted halfway through a 6 mile run. I can't even sleep forever; the body wakes me up when it's hit that particular limit, with no input from me on whether I agree.
I think the thankful antidote or defense to this limited experience and limited mental/brain-i-al resources that all persons experience is the ability to switch off: when I recognize my empathy has reached a limit, I can switch gears from empathizing, perhaps, to trying my best to listen, or learn. When I am out of patience, I can divert my attention to something less trying. When I am out of reason because, perhaps, my emotions are too high -- I can realize this, consciously decide to stop engaging analytically (or to stop allowing myself to turn to 'logic' or overthinking) and practice letting go, or accepting that not everything can be proven, defined, known -- can be mathematic in nature, I'd like to say.
There are limits to empathy, just like any other human trait or ability; knowing what our personal limits are allows us to recognize when we have reached them and to stop trying to wring a dry sponge, or get the car going by flooding the engine. It is useful to know when a given approach may no longer be viable due to resource constraints so that then, we can either admit this limit and step away from the task, or we can admit the limit and begin to try other, potentially also-valuable approaches.