As far as I understand it you are born with your personality as in the preference how you think, make your decisions, perceive your surroundings,... and I do think that does not change much. But parents, the environment etc. have an influence on how you behave to the point that you think the nurture is your nature.
Studies show for example that there is a difference in the brain of introverts and extraverts. But a lot of people here don't know about introversion and extraversion and since the society favours extraverts and children get told by their parents and in school that they have to behave like one. As adults they are still introverts but they behave like extraverts and some of them think that they are extraverts and would probably even answer questions on the MBTI test as if they were one.
I think that a lot of people don't understand that there is a difference between their true preference, what they think should be their preference because they were "trained" like that and their behaviour. For example a female aquaintance of mine did the test, then read the profile and told me that it's simply not her. I told her that the answer is not necessary how she behaves in order to survive in society but how she would prefere to behave if there was no pressure from society. Before she tested as a strong F and afterwards as a strong T. Then she read the profile and was surprised how well it described her way of thinking. I had the same problem with P and J because I was raised in a very organized household and my parents (both J) always told me how important it is to plan (and follow that plan), be organized,... and as a result I behave like that most of the time and when I did the test the first time I tested as a J because I thought that my behaviour is my preference. But I definitely crave flexibility and spontaneity and now that I am living alone and can be more myself I realise how much the learned behaviour is not me.
That is also the reason why I think the Big Five Test is more superficial than the MBTI.