The Sanskrit language is an Indo-European language; this is absolutely certain. Therefore, even if the language itself is native to India, its ancestry is not. However, genetically, there has not been a mass migration into India for tens of thousands of years. Therefore, instead of having mass human migration, we have cultural/linguistic migration, probably similar to that of modern Western culture migrating all over the world...
^ All this.
It is equally as possible the language and culture began in India, and then migrated throughout Indo-europe.
The OIT makes even less sense than the AIT. It makes no sense, nor is there any evidence except for wishful thinking. If, as all evidence suggests, modern humans migrated out of Africa beginning about 150,000 years ago, then migrated north, west and east, it would make no sense to migrate to south Asia, develop a particular language family, then migrate it north and west again. Especially if one considers that IE is hypothesized to be only about 6,000-8,000 years old. No one knows what language(s) early man spoke and when. Because languages change at an alarmingly fast rate, it is not possible to reconstruct them backwards more than 6,000-8,000 years.
Moreover, Neanderthals and modern humans are both descendants of Homo Erectus, which got to Europe and Asia before modern man or Neanderthals evolved, several hundred thousand years ago. H. Erectus has been found all over Europe and the middle east. It's more likely that a pocket of west-central Asian nomads became the Indoeuropeans and radiated outwards. It makes no sense, not to mention no evidence for, that a major language family would have its roots at Point A (India) and migrate to Point B (the Iberian peninsula and Scandinavia).
There is, however, plenty of evidence for a radiation of IE language and culture from a central point. Humans made it to east Asia and Australasia by about 50,000 years ago, but those languages are not even closely related to IE. Why would a language family migrate linearly in one direction, but not another, when it clearly radiates? IE which includes Indo-Iranian, Greek, Balto-Slavic (Russian, Polish, Lithuanian), Germanic (that's actually open for debate), Italic, Celtic does not venture much further than India. Sino-Tibetan and Australasian languages do not go much further west than India and Tibet (except Malagasy, which is Austronesian); Turkic, which includes Mongolian and Turkish, may have a common point of origin somewhere in central Asia.
