sounds like tapeworm segments, but vet should have recognized that.

It might not hurt to give the kitty something for tapeworms.

I wish you the best.

I agree with Sylvia that it sounds like tapeworm segments and your vet should have caught this. This is what I use to de-worm my cat. It is safe and kills all types of intestinal worms including both types of tapes.
http://www.pets-megastore.com.au/wormoutforcatssmalldogs6tabsx5kg11lbs-p-423.html

Had an x-ray done recently which, I'm guessing, should've shown anything internally. The pieces are hard when there, and it takes my finger nail to break them up. Doesn't really sound like a worm to me. Surely they would manifest themselves in another way: stool, maybe on his rectum. Vet has said more than once it wasn't worms. I'll keep an eye on it...

Is kitty eating something around the house, like something plastic-y? Litter?

X-rays don't show up everything, some plastics, especially are translucent to them. Nobody would have suggested tapeworms if you had mentioned they were hard, before the fact. Did your cat see the vet because of frequent vomiting?

It does sound like he ingested something less natural than a bad mouse.

As long as he's getting rid if it & doesn't have any signs of a more serious effect (bloody stool or swollen belly indicating internal lacerations, etc), I'd watch it but not get scared.

I wonder: do dogs have the same impulses that make toddlers swallow coins & stick crayons up their noses?