This is a very interesting question, and I just had to look for the answer. And from my searching, it appears that the answer I've found is no; the genetic difference between African and Asian elephants is too great to allow them to interbreed.
Moreover, according to a BBC News article, there are actually two different species of African elephants: savannah and forest. Researchers have discovered that these two species are incapable of interbreeding as well, stating that the "difference between the two is more than half the genetic distinction between African and Asian elephants."
I said that the answer appears to be that the species cannot interbreed. However, there is evidence of one such pairing: Motty the hybrid. The father was an African elephant, and the mother was an Asian elephant who were kept in captivity at the Chester Zoo in England. Here's a description of him:
The staff had observed several matings between the elephants, but since a cross was thought to be impossible, none expected a delivery. The male calf, named Motty, had an African elephants cheek, ears and back, while nail numbers, (5 front, 4 hind) and the single trunk finger were like Asians. His paternity was determined through immunalogical tests. Sadly, this sensational elephant died two weeks after the birth. It was an early birth and Motty had stomach problems.
Click on the link to hybridelephant.com for a picture of Motty and his mom. No other hybrid birth has been recorded.
I hope this helps.
It would appear that the two kinds of elephants are considered to be different species. Not only that, they are not even from the same genus.
In the wild, of course, the two species would never come in contact since they are from widely separated parts of the world. In captivity, there has been only one know mating between elephants from the two species that has produced a live birth. This happened in 1978 at the Chester Zoo in England.
However, the offspring was not able to survive very long. It died after only two weeks of life.