Both. I used to not be able to sing at all - I was practically tone deaf, since I never had any sort of vocal training. Once I started taking Chinese, I had to improve my vocal abilities quite a bit, since pronunciation is key (there are different tones that require specific vocal inflections). Now I can sing much better, but I'll never be able to hit the high notes that my chorus-trained friends can, and I doubt I'll be able to beatbox decently. A good friend of mine also has practically perfect pitch, which is very much a born-with attribute.
The difference between being born with and learning abilities is why a lot of these actors who try to start singing just suck - despite good coaching, they just weren't born with the ability.