As a child, my very high sensitivity worked intensely even though I might try to defend like mad. No emotion precluded it. Anger, fear, or intimidation did not shut off my sensitivity or cover it over as many people commonly could do. My sensitivity worked overtime no matter what else happened.
The only thing that did work was denial. To use denial to deal with grief I had to throw a very gross "big switch" that cut me off from everyone and everything that used to bring me pleasure. Later I was wary after that of doing anything but experience however I felt, no matter if it was in intense high definition.
Therefore, I imagine that certain people are born more sensitive than others, but sometimes I believe that we all have high sensitivity and some supress it. Many are tempted to deliberately use other emotions to shut down their sensititivity as a covering up strategy. For instance, feeling angry feels more powerful than feeling sad. After a habit is designed and in place, one emotion precludes or dull natural sensitivity. It takes much effort to stop because the habit becomes an assumption that works insidiously. From what I understand, sensitivity for most others needs safety to emerge or for them to notice it behind other more savory emotions that are underneath their other more intense unsavory emotions. However, if you merely remove the habit, natural sensitivity will rebound as if by itself.
Perhaps because of my high sensitivity to both external events and my internal responses, I was sort of "forced" into experiencing other people's emotions. I did not learn that I was experiencing the emotions of others until my teens, when gradually I learned that my emotional empathy seemed to be controllable by proximity.
It took much practice before someone could ask me how I felt and I could answer truthfully. In the interim of learning the skill of separating out other's emotions from my own, I sometimes needed to turn away and not look at them while I was asking myself what I felt. At the beginning I needed actual physical isolation to sense myself as being distinct from others. Now I'm able to carry a sense of autonomy with me, knowing where emotion is coming from usually with certainty. It was a long road!