SAN MATEO, Calif.—For three months this summer and fall, in an open-plan office here stocked with Red Bull and ramen noodles, Brian Gamido and a couple dozen other true believers toiled at what he conceded sounds like “a pipe dream”—building viable businesses based on bitcoin.

But this bitcoin boot camp was hardly a shoestring effort: Mr. Gamido and the other fledgling entrepreneurs were bankrolled in part by one of Silicon Valley’s most storied families, which has emerged as among the alternative currency’s biggest supporters.

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