Well (almost) anything is possible...
Welcome to the RTA Forum!
Driving from the US into Central America isn't that much of a problem. There are a number of legal headaches associated with driving in Mexico and points south, but they all can be overcome with some patients and planning.
The real problem would be continuing south into South America and overcoming the "Darien Gap." There is no road that runs from Panama into Columbia, so you need to find a ferry that will ship your car across the water.
The Driving to South America Thread, which is the first thread you'll see in the off-shore roadtrip section, has some good first hand advice from people who have actually done this trip.
For inspiration, I also highly recommend reading Road Fever, a book detailing a record setting drive from Argentina to Alaska. Its a great read that will certainly help keep you in the mood as you plan out your trip.