Hi there, this is Dr. Elaine.
How long has this been going on?
What is her diet?
Is she an egg-layer?
Any history of illness?
Drinking a lot of water?
I would be concerned about a couple of possibilities here.
Diabetes occurs in budgies, and as it can in other species, adversely affects circulation to the extremities. this can cause a gangrenous situation within the claws, whereby they die and slough off.
Budgies are also prone to tumors. Some are associated with the kidneys or the left ovary (the only remaining ovary in birds), and by virtue of their location can interfere with blood supply to the feet.
The claws in either case are the first to suffer, as they are the furthest out in circulation.
Some cases of diabetes in birds are actually secondary to infection, so it is important to have her seen if possible.
A very treatable problem would be mites (scaly leg mites)--these also cause claws to overgrow, split fall off (not necessarily turn black). Ivermectin (200ug/kg) can treat (orally). Be CAREFUL with dosages--check with a veterinarian.
Hopefully this sheds some light on matters.