Are leaders born to lead or do they learn to lead?

In my opinion, leaders are made, not born. There are some personality characteristics that are good to have to be a leader, but leadership is largely situational, which implies that different contexts require different kinds of leadership behaviors.  Let's look at a few different contexts to see how this bears out.
Some people say a good leader should be entrepreneurial. That is likely to be true in some situations, such as high-tech start ups. Being entrepreneurial implies having a risk-taking personality, though, and that does not always make for a good leader. There are many organizations who want someone risk-averse. No matter what kind of personality a leader has, he or she will have to learn to adjust to different industries, cultures, and economic times.
Some people are autocratic, rather than democratic, in their styles and personalities. If all that is needed is to keep things moving along on a production floor, that might be the best way for a leader to behave. in a creative endeavor, however, that style simply will not work. A leader must be able to discern the difference and understand how to motivate people to do their jobs. 
Some people seem to be born more introverted than extroverted. If they cannot adapt, they are not necessarily going to be successful in a sales-driven company or in a company in which they are expected to be the public "face" of the company with the expectations of being responsible for fund drives or press conferences. People can adapt, though; they can learn to do what is necessary to lead. 
Being born with a certain amount of flexibility and resilience is probably vital to a leader's success, since leadership today requires virtual shape-shifting, the ability to adjust one's style, and the ability to roll with the punches that modern-day business brings.