There are several problems with Stevenson's method. He often worked with translators in countries about which he knew very little. Questioning anybody is tricky, but questioning children is especially tricky. "Interviewer bias is the central driving force in the creation of suggestive interviews" (Bruck, Ceci, and Helmsbrooke 1998; quoted in Mills and Lyon: 303). Questioning children and adults via a translator introduces another element of uncertainty regarding the bias of the questioning technique. Most of the interviews took place in countries where reincarnation is an accepted belief. So, the translator would be "typically imbued with the cultural expectations that past-life recall is a valid phenomenon" (Mills and Lynn: 303). Stevenson, being non-fluent in the language and the culture, was in no position to assess the reliability of the questioning by the translator.
There is also the obvious problem of confirmation bias. The ideal, according to Stevenson, was to seek out PLE stories and then try to confirm them. Failure to confirm, however, did not count against the reincarnation hypothesis. In fact, nothing could be discovered using Stevenson's methods that could ever disconfirm the reincarnation hypothesis. Many scientists would consider this a fatal flaw in his methodology.
Another problem is that there seem to be alternative, non-paranormal, explanations for all of his data. Stevenson was aware of the fact that many of the features he was detailing were culturally driven. He wrote:
Critics of the cases have therefore suggested that a child's fantasies, perhaps of an imaginary playmate, may become shaped by its parents and peers, through their questions and suggestions, until the child assumes an identification with a deceased person. In this way the child becomes the subject of a factitious case suggestive of reincarnation.
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Coincidence, faulty investigation, deception, and other normal explanations are available. "Wilson (1982) proposed that people reporting PLEs are motivated by a desire to identify with a higher social class" (Mills and Lynn: 294). This concern seems especially relevant when dealing with cases in India and Sri Lanka. Sometimes cases Stevenson considered "solved", when examined by others, turn out to be less than pristine. For example, Stevenson found many claims by Sunil Dutt Saxena of Bareilly that matched events in the life of Seth Sri Krishna of Budaun. Both cities are in northern India. Ian Wilson notes, however, that a local doctor had explained to Stevenson that Sunil had been coached by Sheveti Prasad about the details regarding Krishna, whose family rejected Sunil as the reincarnation of their relative (Kelly: 91; Wilson 1989). Stevenson rejected the evidence against his case and considered it "solved."
http://www.skepdic.com/stevenson.html