(Virginia Woolf: A writer who was quite preoccupied with sensibility in her various writings.)
As originally understood in the 17th century, sensibility starts out as sense information taken in and/or expressed. But sensibility was also thought of as refined experience, judgement, and truthful expression. There has always been an authenticity about it in which the practitioner is true to him/herself. Fine taste is also associated with the term with an emphasis on purity and fineness of feelings, sensations, and experiences.
And yet the latter does not mean that sensibility is mere snobbery; rather empathy and sympathy are part and parcel of its experience and expression. It is, above all, an inwardness tending toward the best living of oneself, as in more ennobling or elevated feelings and resulting moral affects.
The person with sensibility is, above all, very much in touch with his or her feelings, and often engaged in the process of making one’s own soul at a higher level.