Hmm, interesting question.
If you gave someone with a natural gift and someone who didn't have it the exact same training, and both worked hard, the gifted person would most likely turn out better. However, the other person could rise to match the gifted's skill.
Allow me an example...I'm in two choirs in the College of Fine Arts. I had to audition to get into them, and when I sang at the first rehearsal, I thought '...how the CRAP did I get into this?' because most of the people in it are voice majors or music majors and they're all fantastic singers and they can all sight-read music ridiculously well.
Over time, I myself have learned a lot and gotten tons better in both singing and sight-reading...however, I still see a massive gap. Why? Because I don't have access to a vocal teacher...and my CFA Voice major friends sing for multiple hours every day and have voice lessons. So while I can get better on my own, and possibly with my old teacher during the summer, they are inevitably going to outshine me in absolutely everything because it's what they do all the time.
Another example is when I struggled to keep our tiny choir together in high school...two of my best friends were in it and they really wanted to be there...but one had absolutely no sense of pitch. So when there's six of us, off-pitch notes are extremely noticeable. I tried working with her to help her hear the notes better (and she was for it at first), but she didn't bother trying and just walked out because she didn't want to put the work in.
So, not only did she not have a gift for it, she also didn't want to work at it to get better. As opposed to a friend of mine in my theatre group who also was tone-deaf when she started but worked hard for two years with my voice teacher and now can sing on-pitch wonderfully. She's not a shining star, but she's good because she worked at it.
In conclusion, I agree with Biff that practice makes perfect, but you have to want to do it, and also have at least SOME talent as well. Practicing mechanically doesn't get you anywhere. Practicing with drive and passion gets you to the finish line.