Don’t they? Come on, everybody has feelings, don’t they?
I believe that the vast majority of people (if not everybody) is born with a healthy and balanced emotional system.
What happens afterwards, maybe from day one and onwards, is the disturbance of this balance due to one’s encounter with perceived part of life experiences (not all experiences have an effect on one’s psyche). This contributes to shaping one’s character, also in terms of social and emotional interaction. Such disturbance of natural emotional balance may cause the increase of emotional sensitivity/awareness or the (gradual) numbing of it, sometimes as a mere defense mechanism or traumatic reaction to an event or a series of events, I believe.
But, yes, I do believe that some people may reach the level of a totally numbed down (shut down) emotional system. I believe psychopaths have such a thing as they’re characterized by having total or major absence of conscience or the feeling of guilt.