I installed a Lokar oil dipstick not too long ago. I went to check the oil yesterday and when I pulled the dipstick tube out, much to my surprise the gauge section of the inner rod on was completely broke off.

What is the worst thing that could happen still driving the car? Could the oil pump suck the piece up and lock up? Give me your idea on worst case scenario and what you think is likely to happen.

 

The pick-up only has a course screen over it so yes very small pieces can get picked up by the pump. However most of the big pieces will sit on the bottom of the oil pan. The worst would be the tub gets picked up by the crank and beat to death, leaving lots of metal in the oil which can eventually get into the bearings and rings and scratch the s..t out of everything.
Oh and if you did get a big enough piece into the pump it could jam and shear the oil pump drive.