Personally, I don't think of it as empathy, but more as a sense of duty, like "Here's someone who needs help, I'm in a position where I can help. Sure, I can ignore her problem, leave it on the backburner while I clear the Q and make the richest company in the world a couple bucks richer, but why not do something that actually has a tangible benefit?".
Now, if this person is an ass, I wouldn't wanna go the extra mile, though I might, just to spare the unpleasantness from coworkers every once in a while. Call it human nature, but its easier to go above and beyond the call of duty when you're approached with honey. Either way though, if after doing my part (whatever that may be), the situation isn't resolved, I simply let the patient know. If it's insurance rules, I tell them that. If it's something with company policy, I tell them that. If their doctor screwed up, I tell them that too.
Of course, it helps to apologize for the situation, even if you had nothing to do with it, and at least to verbally empathize, even if you don't really feel it. One method I've found really helps to diffuse a situation is to place yourself on their side, I.E. saying stuff like "believe me, if it were up to me, I'd do this or that or the other thing", or "I hate to say it, but our computer system can be dumb sometimes, and sadly stuff falls through the cracks..." or especially this one: "Sadly, that's correct, insurance companies are a beast to navigate around, we hate every second of it too". Personally, it's much easier to actually believe lines like this, as opposed to acting shocked when someone's copay from last year shot up $20.