A difficult first-book decision
The archived author biography says Christopher Little spent a year trying to place the manuscript before Bloomsbury offered in August 1996. Editors evaluating it then faced an unknown author and a publishing commitment; the later readership of the series was not evidence available in the submission pile.
A count with uneven paperwork
Later reporting commonly says twelve publishers passed, and Rowling has described receiving many rejections. The broad path from repeated refusal to Bloomsbury is secure, while the reasons and surviving documentation for each individual pass are not equally visible—an important distinction from the polished legend.