The only obvious way to shear off the tip that I know is if you reinserted the stylet while having trouble threading the catheter, puncturing and shearing off the catheter. That fragment can potentially float downstream--medical term is an embolism. Wherever it gets hung up determines the problem--heart valve, lung, brain, etc.
I can't picture in my mind what is missing and presumed inside the patient. I mean, when I discontinue an IV, I always eyeball the catheter to verify it is the expected length, smooth tipped and intact. What could possibly be lost in the bed??? I mean, if you remove the catheter and it's not intact, it's in the patient. They don't just crumble!
I would guess that flouroscopy would be the obvious tool to look for the thing. Most insertible medical devices have a radiopaque stripe so you check for placement.