Music Training: Can you teach musicality?

We all think about, dream about, and talk about being more than we currently are. More talented, more handsome, taller and generally more. One of the areas that people are always looking to improve themselves is to be more talented; whether it be better at drawing, or painting, or dancing or, and perhaps the most common, is to be better musically. People are always telling me that they wish they could be better singers, or learn to play the guitar or the piano. But can you teach these kinds of skills via training? I know that sounds strange because you can take lessons in singing, or guitar playing but what I’m talking about is musicality.
People talk about having perfect pitch or being a “natural” musician. You can read up, you can practice, you can learn and you can improve. But there is no replacement for natural, musical talent unfortunately. Some people will always be better.