Some people can be turned into leaders. I have seen people really change when they were given some direction. However, some people seem to be natural leaders. I think that the people who appear to be natural leaders are given opportunities to lead and learn to lead early in life, and have natural personality traits that lend themselves to leadership.
Not everyone has the basic raw material to be a leader; some people are too indecisive, too dishonest, or simply lack the ambition to lead. Of those who do have the requisite inborn leadership qualities, some will become leaders because the opportunity arises where they can, or sometimes even must, lead. Others may have the qualities but never get the chance to lead.
On the counter argument side, I think that most people would agree that a person who is a hero is, by definition, a leader of sorts. There are many people who become heroes because they find themselves in circumstances where they act without thinking; in many cases these people are folks who would never have become a leader in any other way.
My answer is "yes." Leaders are both born and made.
There is no doubt that some people are born with more charisma and more of the stuff that leaders are made of. However, leaders are also formed by their circumstances. There are surely any number of people who are born with charisma but who are never put in a position to demonstrate that and turn it to leadership. In this way, leaders are made by their circumstances.
Leaders are made, not born, with the sole exceptions, to the best of my knowledge, being leaders of various religions in which the faithful subscribe to the idea that the leader was predetermined somehow, such as the Delai Lama, who is believed to be "born" by adherents of Tibetan Buddhism. Research strongly suggests that we are born with a tendency in some personality attributes, for example, shyness or risk-taking. We might also be born with various tendencies to "intelligences," such as musicality or logic. However, these are all tendencies, genetic combinations that may or may not be switched on by our environment, a process we call epigenetic. Even a person who is born with a good combination of leadership traits may or may not be subjected to an environment in which those traits are encouraged or called upon in some way. Additionally, what makes a good leader is often a question of context, with various times and places needing different kinds of leaders, sometimes a risk-taker, sometimes a steady hand at the helm. This implies that a born leader somehow magically finds his or her proper context, which is wildly unlikely to be the case in most instances. So, all in all, leaders are made, born with a collection of traits that may or may not be forged by the world around them, that may or may not called upon in the context in which they could be of use as leaders. A leader who is born is one whom people believe was born, logically implying that this is a question of faith, not a question of evidence supporting the belief.