Thanks for writing. The roots on a new orchid pup (called a keiki) need to be at least 2 inches long before you separate the plant and pot it up. So it may be that you're cutting it off too early?
Also: When you do pot it up on its own, be sure you use a fine-grade orchid mix or the moss. There's a better chance of the plant not making it if you use the traditional, coarser mixes.
Keep your baby orchid well watered, but not wet, and it should take.
---Justin, Senior Garden Editor, BHG.com