Paul Bragiel worked for 15 years to become a Silicon Valley player. Then one morning this January, the venture capitalist woke up at a Berlin tech conference with a lingering cocktail-party buzz and this preposterous thought:

What would it take for a self-described "chunky, out-of-shape computer nerd" to make it into the Olympics?

It's the kind of fantasy many people have entertained as they watch the games, inspired, and then have quickly dismissed as irrational.

But what Mr. Bragiel did a week later shows what can come of Silicon Valley's unfettered confidence, analytical bent and network of personal connections. He put his career on hold to pursue a spot in February's Winter Olympics in Sochi, Russia.

He could afford to fund the dream. And as a startup founder, he says, he's by nature "irrationally optimistic."