Do you have to be in a management role to be a leader? and explain

If so, and assuming that leaders do not make good managers, does this mean that in order to have good leaders, you must accept bad managers?

If this is not so, is it possible to groom good managers from the ranks of non-managers that show good leadership?

And finally, if the assumption is that good managers are generally not good leaders. Is it possible to make managers good leaders? In other words, can leadership be taught?