Yes, there are many cases where the body's immune system can completely cure a modest cancer infection. I believe the common term used by doctors is "the tumor magically melted away". However, if the immune system leaves just 2-3 tumor cells left that are not targeted, they can grow back into a full tumor.
When the immune system learns to destroy cells, it also remembers the markers for bad cells. If you have (for example) cancerous liver cells and your immune system learns to destroy these, the potential exists for it to learn markers that are also present on normal liver cells.
After your cancer is gone, this memory that partially (or entirely) applies to regular cells will lead to your immune system trying to destroy normal, healthy cells.
