I'd read before that Stanley Ann Dunham, as a minor at the time of Obama's birth who couldn't satisfy the residency requirement under then-existing law, was ineligible to transfer American citizenship to her child. From his mother's side, Obama is, in effect, a "stateless person." On his father's side, he is a British citizen. But as a further complication, his British citizenship may have been terminated by his Indonesian step-father; in which case, he'd be an Indonesian naturalized citizen.
On these facts, it is certain to me that the case cannot be made that BHO is a natural-born American citizen.