Usually the first problem is EGT's, until you get the turbo's compressor and exducer balanced for your power level, or use water/meth injection. Then it's head gaskets, and next it's head bolts and sometimes head cracks, depending on which head you're running. 7mm injector heads are a little tougher than 9mm injector heads, but you can blow either one clean off the block if you try hard enough.
Turning up the VE pump too high can blow pump seals etc. and a piston lift pump can do the same if it's the wrong pump or is running too high of pressure. Still, with all of that considered, you can get a whole bunch more power out of one with no real harm, except maybe shortened engine life overall.
I'm assuming the original question is in reference to a street engine, rather than a puller or drag race truck. Very few street machines are set up to exceed about 3,200-3,500 rpm. Certainly another factor is basic engine condition: brand new, or fresh rebuild, or low mileage used, or high mileage used? If you want the engine to last it's built one way, and if you're shooting for the moon on power it's built another way. I think maybe Charles was more referencing moon shot stuff.
A stock Cummins in good condition can easily tolerate performance modifications that double its stock power output with no component failures if driven sensibly. Stock power is 105-120hp, so 240hp will still hold together just fine, however engine longevity will suffer if it's continuously driven hard, just like any other engine will fail. On average the Cummins outlasts all the others for durability, and that's why we're all here!