Can cats get brittle claws?

it's normal.
Cats claws grow permanent, and they grow in "layers" to ensure a cat has always sharp claws which are necessary to climb and to defense themselves against predators and other bigger animals which could be dangerous for the cat.

Clipping a cat's claw is like cutting off the tip of an onion.
The outer layers are destroyed and will fell off (or easy to peel off the onion) after a while until the next intact layer is reached.

If your cat is using her scratch poles only I would say you could stop clipping for a while or even for ever.

Yes a cat can get brittle claws / you may want to get them some different cat food that supports their hair and nail growth/ / your finding cat claw peices around could be just normal nail sheddings too/ and yes maybe try stop clipping for a while to see what happens / dont worry its not a life threatening thing /

The claws are sheathing. This can happen more often if you clip them with conventional nail clippers. This is because cat claw clippers or people nail clippers split the nail often times, causing the outsides to get weakened and fall off. This also happens naturally since cats claws grow a lot and are retracted therefore don't get a god chance to wear down. It's to keep them from growing too long and harming the paw. Clip their claws closer to the edge rather than close to the quick, which will lower the damage to their claws, or maybe invest in a Pedi Paws. This machine files the claws rather than clips them which is much better for their nails. Some cats are afraid of these though. They may also have a nutrient deficiency which is causing this, so try and rule that out by visiting your vet and closely examining the foods you are feeding them. They should be on a senior diet now, go to the vet more often, etc. Good luck!