Apparently its not completely possible. This lengthy but interesting (and pic filled) story from Dark Roasted Blend details the many many reasons why attempting such a trip is not only very challenging but nearly suicidal.
A section of Southern Panama called the Darien Gap is not only roadless but filled with all kinds of impedance such as bandits, critters, poor terrain, unnavigable streams among many other things generally considered unfavorable for travel.
Check out an excerpt:
“If you ever wondered what’s like to be kidnapped in the Darien Gap by Colombian Death Squad, here is an account:
National Geographic Adventure contributing editor Robert Young Pelton and two American backpackers were kidnapped there in 2003 and held for 10 days; here is what he has to say (once they got out of there alive) –
‘The Darien Gap is one of the last—not only unexplored—but one of the last places people really hesitate to venture to… It’s also one of the most rugged places. The basic problem of the Darien Gap is that it’s one of the toughest hikes there is. It’s an absolute pristine jungle but it’s got some nasty sections with thorns, wasps, snakes, thieves, criminals, you name it. Everything that’s bad for you is in there.'”
(via Dark Roasted Blend)