I'm not poking fun. I actually posted something up asking the same question. I put it under how to have more control when singing or something like that. I didn't know what it was called. I asked how to sing ad libs because I thought that was what it was called. lol. I actually have more questions about pop and r&b singing than classical music.
It can be trained, I believe. But there are limitations, of course, depending on how good your ear is in detecting what's in tune and what is not, and the agility of your vocal chords. People with lighter voices can do this more easily because of their thinner vocal chords, which translates to better agility. This is why dramatic coloratura sopranos come in limited edition. :-) Hope this helps.
There are vocal exercises you can do 2 get the flexibility & accuracy needed to do melismas. You should be able to get a (book or other) recommendation from a local voice teacher. If you haven't already thought of it, practice the melismatic parts of your song separately, then work them in. Hope that helps