FORTUNE — In retrospect, the biggest blunders often seem inexplicable. Four different book publishers, for instance, passed on J.K. Rowling’s first Harry Potter novel. A weird story about the adventures of a juvenile wizard and his friends just didn’t seem worth a $5,000 advance. Oops.

According to author George Anders, a Pulitzer Prize-winning editor at The Wall Street Journal, most big companies make comparable mistakes all the time. For a new book, The Rare Find: Spotting Exceptional Talent Before Everyone Else, Anders set out to analyze how some of the most successful enterprises choose extraordinary new hires.

What he found will come as no surprise to anyone who has worked with someone who looked good on paper but turned out to be less than stellar in action.

Instead of insisting on a rigid set of credentials, Anders says, hiring managers ought to focus on what the job really requires and give a fair shot to candidates whose resumes may be what Anders calls “jagged,” or full of ups and downs.

Someone whose background “appears to teeter on the edge between success and failure,” he writes, can do “spectacular work in the right settings, where their strengths dramatically outweigh their flaws.”

Consider, for example, legendary Facebook engineer Evan Priestley. He had changed his college major three times before dropping out altogether, and was working as a low-level web designer at a small firm in Portland, Me., when, in 2007, he happened to come across a programming puzzle that Facebook had put out over the Internet. Priestley’s solution was so elegant that Facebook flew him to Palo Alto for an interview, where he impressed everyone with his skills.

Facebook hired him, and the rest is legend: Priestley led a team of programmers that sped up Facebook’s infrastructure and made it easier to add games, maps, and other applications.

At one point, Facebook’s site stopped working for a small group of users who, it turned out, were hampered by an obscure, out-of-date security program. The only publicly available manual was written in Danish. No problem! Priestley and a coworker stayed up all night learning enough Danish — mastering terms like foutmelding and beveilaging — to untangle the trouble.

The point: If hiring managers had considered only Priestley’s lackluster resume, he’d never have gotten a foot in the door.

Drawing on other case studies from organizations as diverse as the FBI, the National Basketball Association, General Electric (GE), and (a cautionary tale) Enron, this is thought-provoking stuff for anybody who’s frustrated with trying to find exceptional talent using the same, tired old methods. The Rare Find is also a rare find in itself: A business book that is actually fun to read.