I have a US passport and an Indian 10 year visa in my old US passport. The visa expires on Aug 19, 2015. Can I travel in India between May and June 2015? or do I have to have at least 6 months left on the visa?
It depends - the visa, not so much, but when your US passport expires.
Yes, a visa is valid until the visa expiry date. So in your case, you can arrive in India on August 19, and leave again the same day if you want - your visa is still valid. So travelling there between May and June is fine as well, as it's still valid.
So no, it doesn't matter how long is left on your visa, however - you need both passports, AND your new passport must have enough pages left for any stamps enroute (one or two is usually enough), and at least six months of validity on it. But that doesn't affect too many people.