Are there reliable accounts of Buddhists who have remembered past lives? Although the vast majority of people will not remember past lives, are there reliable accounts of people who have? Specifically are there accounts of Buddhists who have remembered past lives and whose experience can be (even partially) verified in some independent way? Or do people with past life memories generally come from other religions and traditions such as Hinduism?

I appreciate that reliable is a very ambiguous term. If possible could people perhaps note the quality of the evidence when answering. Is it anecdotal, statistical significant etc...?
Dr. Ian Stevenson did research on rebirth, and most of his subjects were children. As I remember he said during the first 5-6 years of a child's life some of them would remember how they died. Most of them who remembered their previous life have had some sudden tragic death. Hence, these incidents seem to be deeply carved in their minds, but this memory would slowly fade away as they grew old. 

Most of these incidents were reported from families having Hindu or Buddhist backgrounds because their parents believed in rebirth and believed their child was saying the truth. In some cases, when the researchers traced the child's past routes based on their stories, they had actually found that the person had existed.  Also in some cases it turned out that the parents have been lying to gain some popularity from local media. :)

But I think this research is not so popular among other scientists, claiming it has no clear evidence that backs his theories. Upon his death I think his student continues this work. 

Here's link to their work. 
http://www.medicine.virginia.edu/clinical/departments/psychiatry/sections/cspp/dops/publications-page
(Edited the previous reference according to Ven. Yuttadhammo's comment)