Firstly it's not a palm, not even close. It's Dracaena marginata (more closely related to asparagus than to palms). But they should put out new shoots from those stems. You need to be sure the soil drains fast and doesn't stay too soggy. Don't over water, let it dry out a bit between waterings. You can push a wooden dowel down into the soil to check how wet it is deeper down. They like a lot of light, but ease them into direct sun if they've been growing in shade. The broken off branches can be used as cuttings, they strike quite easily.

Such great news!! Thank you for your wonderful feedback. I've got hope :D

Cats are good plant killers. I had a few kill a large outdoor Schefflera by a thousand paper ( claw) cuts in the trunk climbing it. You might wrap something around the top of the pot to discourage the cats..a bit of chicken wire...something. Be creative!