The simple answer is no - you cannot travel overland between Panama and Colombia, or vice versa. There is no road and no public transportation - the Pan American highway does not run through the Darien Gap region of Panama.
The Darien Gap is huge region of primary rainforest that is both dangerous (due to the presence of guerrilla groups and drug traffickers) and almost completely impassable. If you had a guide, a security entourage and a humvee or motorbike, you might just be able to pass through the Darien Gap from Panama to Colombia, but almost certainly you have none of these, so here are some suggestions of how you can travel between Panama and Colombia.
Flights & Airpasses
The simplest option is to catch a flight from Panama City to either Cali, Medellin, Cartagena or Bogota. Try booking these flights with the airlines Copa, Avianca, Aerorepublica and Aires - they're really not so expensive. Aires is normally the cheapest - see the flights to Colombia section for more tips. If you have travelled from outside of the Americas to Latin America (eg. you're visiting from Europe), you should book these flights as an airpass as it's much cheaper than standard tickets. The type of airpass you want to book is called an "All America Airpass" (alternatively called a Hahn airpass).
Capurgana to Puerto Obaldia by scheduled boat
An alternative, cheaper and more adventurous route is as follows (described as if one is travelling from Colombia overland to Panama). Travel to Capurgana, which is on Colombia's Caribbean coast, close to the Panamanian border. To get to Capurgana you'll either need to catch a flight from Medellin, or you can travel overland by bus to Turbo (from either Medellin or Cartagena - check the current safety situation), from where it's a two hour boat trip (about $25) to Capurgana. There's some fantastic beaches in Capurgana so you'll probably want to spend a couple of nights. From Capurgana, you can catch a boat (relatively scheduled services for about $20 per person) for one hour to Puerto Obaldia in Panama. From Puerto Obaldia there are flights to Panama City with Air Panama (flights are about every other day). If you prefer not to fly, instead you can catch an onwards boat from Puerto Obaldia to Mulatupo or El Porvenir (both in the San Blas Islands - Mulatupo is much closer) and pick up a flight to Panama City from either, or alternatively take a boat to Miramar, from where you can take a bus to the major port city of Colon and another on to Panama City or wherever.
Travel from Capurgana via the San Blas Islands to Panama
From Capurgana, you should also be able to arrange a private chartered boat to take you to the San Blas Islands of Panama, from where you should be able to charter further boats through the San Blas islands until you reach one of the various airports there such as Mamitupo airport. This is quite a long route, and it's unlikely to be cheap unless you get lucky and hitch some rides, but it is extremely beautiful. Such trips can be organized by the Darien Gapster.
Cartagena to Panama & San Blas by Sailing Boat
It's increasingly popular to take a sailing tour from Cartagena through the beautiful San Blas Islands to Panama. This sailing trip is a very scenic 5 day tour - sailing boats usually depart when they have enough passengers wanting to travel. The cost is approximately $350 per person including food and board. Hostel Viena in Cartagena is well known as being a good source of information about skippers who offer such tours. You could try the German operated Seeadler boat (the only boat with a dedicated website). Be warned that the boats that sail from Cartagena to Panama are of highly variable standards, and there are many reports of disasterous experiences, poor quality sleeping accommodation, crazy captains and even shipwrecks. Booking in advance is very difficult as there are no real schedules, and you run this risk of finding yourself with a dodgy skipper - it's best to arrange such a sailing tour locally when you've scoped out the boat and it's skipper. If looking for a boat in Panama, I understand that Hostel Portobello can help travellers with this journey to Colombia.
From Bahia Solano up the Pacific Coast by Boat to Jaque
Bahia Solano is a resort town well known for it's deep sea fishing on Colombia's Pacific Coast. You can fly or travel by cargo boat from Buenaventura to get here. From Bahia Solano, there is an infrequent boat service up the Pacific coast to Jaque in Panama. Boats leave when they're full (or when they feel like it) - the cost is approximately 150,000 pesos per person, and the journey time likely to be around 10 hours. Make sure you get your passport stamped (out) in Bahia Solano if you make this trip - and be prepared to be able to show plenty of cash (plus a yellow fever certificate) when you arrive in Panama (ie. cash to demonstrate you have the means to travel onwards from Panama). During the whale watching season (June to September) you'll spot whales the whole journey. Once in Jaque, there's flights (not daily) with both Air Panama and Copa to Panama City.