So, knowing what I've written, you ask yourself, "Self, if the 360 is baking its processors to the point that they'd separate, why aren't the heatsinks holding them tightly to the main board?" That's a very bright observation!
The problem is pictured below (the second image, silly). Those two X-shaped pieces of metal are what try to hold the heatsinks onto the CPU and GPU. Problem is, they aren't springy enough to do the job. The heatsinks have a tiny amount of wiggle room and the motherboard is free to warp from heat. That prohibits proper heat dissipation and allows the processors to break away from their connections. I blame bad design from Microsoft's need to get the 360 out before the PlayStation 3. But now isn't the time for blame. I bet you're about ready to buy one of those Blu-Ray players that come with gaming functionality.
So now we've determined that those "X-Clamps" need to go. The next step lists parts needed.