The answer to that question, in my opinion and experience, is BOTH. Indeed some people are born with acute psychic sensing. Those who are lucky to be in a family where such ability is inherited, or at least recognized and nurtured, learn from an early age how to handle it responsibly.
In my experience, born psychics are fairly common, while hereditary psychics are relatively rare.
There are many who are born with some psychic ability, but such a thing is beyond the belief system of the family. In these cases the abilities are discouraged and even punished.
As a result the child learns to suppress and fear their talent. Or, they may use their gifts irresponsibly to manipulate others for their own gratification.
Then there are those who are emotionally sensitive or empathic, or who have vivid imaginations and easily see cause and effect in some certain situations. They don't think of themselves as having psychic ability. Usually they think they can't conjure up any psychic talent on demand.
But they may be aware that there are times when answers or knowledge just comes to them, unbidden.
Of course there are the skeptics who think that any claims to be psychic are a scam. Some of these skeptics may have latent psychic talents, yet instead of investigating what their "strange and unusual experiences" might be all about, they fall into the trap of modern society in thinking they must be going crazy.
Unfortunately, such people can end up getting diagnosed with a mental illness unnecessarily.
And finally, yes, there are many people who may be attracted to the idea of psychic powers, but don't have any innate talent, or who just can't develop any skill.
Can You Develop Psychic Abilities?
Indeed, many can. And there are plenty of books on Amazon to use as a starting point, or online courses, if you don't have a legitimate teacher available in your community.
Discover Your Psychic Type is a useful book for absolute beginners. And I recommend The Witch's Shield: Protection Magick and Psychic Self Defense as a very useful resource for anyone who is opening themselves to psychic energies.
If you prefer an interactive course with a group of learners, check out the Viva Institute's psychic development offering with Leigh Hopkins and Echo Bodine.
I also recommend getting into the habit of mindfulness (Vipassana) meditation as a way to become more aware of and trusting in your own bodily-felt communications. Learning this style of meditation has been invaluable to me in developing my own psychic abilities. This kind of meditation will help you be able to distinguish between thoughts, feelings, and sensations that belong to you, and those that are coming to you from the psychic realm.
My favorite mindfulness teacher is Shenzen Young, who has many resources available free online. Check out Five Ways to Know Yourself, along with his many retreat options -- some even by phone or at home.