First of all, there has been conflict. Many South American countries harbor long-standing political and territorial grudges, some going back a century or more. However, the territorial borders between the countries have been stable since the late 1800's, so that in and of itself lends to skirmishes rather than all-out war. As recently as the 1970's though, war was certainly a possibility in the area.

Venezuela/Colombia is a well-known low-level conflict. Chile/Peru/Uraguay is another. It has not boiled over into much other than small border skirmishes, which thankfully have died down since the 1960's but are still ongoing.

Chile/Argentina were very close to war in the mid 1970's. Only the threat of direct American intervention and the implied threat (confirmed decades later) that Chile had stockpiles of chemical weapons supplied by the U.S. prevented it from boiling over into actual war.

The 1970's also saw "Operation Condor", a CIA backed program that created an informal and loose coalition amongst miltary dictators in the area. Though the immediate result was widespread repression and increase in Human Rights abuses, it conversely kept the military heads of state in close contact with each other which kept border tensions low.

South America today is very much like the U.S.-China. The various countries have become so economically dependent on each other that all-out hostilities would affect both aggressor and defender in equal measure. That doesn't mean war is impossible, but it becomes more and more unlikely as their economies become more interlocked with each other.