The case of the Chilean Patagonia is an interesting one. Chile used to be the owner of almost all of the Patagonia region. Little by little, Argentina gained control of more of la pampa. One of the most tense times was the almost-war or quasi-war in 1978 between the two countries under dictatorships. Wikipedia refers to it as the Beagle Conflict (although I haven't heard anyone say that here). Argentina claimed ownership of three small islands called Picton, Lennox, and Nueva, which had always been under Chilean control. War itself never broke out, but troops were guarding the line on either side in the cold southern desert. In the end, Chile kept control of the islands. In my intensive language/cultural class, we watched an exquisitely-done movie about this conflict called Mi Mejor Enemigo (My Best Enemy). I really recommend it if you have the chance to see it!
Bolivian-Chilean border issues are another story. The War of the Pacific in 1879-1883 was a fight for the area that is now Northern Chile. Read the full details in the link if you'd like! In the end, Chile won. Treaties were signed, and Bolivia lost its access to the sea. Peru lost some of its southern territory, including the cities of Iquique and Arica. Here is a map of the territories before the war:
Since 1883, the area that Chile gained has been Chilean land, full of people with Chilean identity that developed for more than a century. However, the current Bolivian government is asking for negotiations about getting the land and their coast back.
All the Chileans that I've talked to about this (3 or 4), laugh it off as a ridiculous request. They say that since the land has been Chilean for so long, there is no going back. And on top of that, they have the treaties with the Bolivian signatures! Chile's president, Piñera, has declared that Northern Chile is absolutely not under negotiation. People are glad about this (in fact, this is the only positive thing I've heard about Piñera). However, my host mom thinks that eventually we will give the territory back to Bolivia because the matter will never be dropped - just hopefully not in her lifetime.
Of course, I am getting a solely Chilean view. I'm sure its horrible for Bolivia to not have ports and sea access. I'm sure there are many arguments on their side for the fairness of returning the land. I'll have to go to Bolivia and find out.