Do you claws have tears or rips in them??? Can you see inside of the nail, to where the quick is??? It is not normal to have bloody nails, unless some trauma has occurred. If it continues, take her to the vet.
Another tip to anybody with animals in the house: be careful using Swifter or other chemicals on your carpets or floors. If the floor is still wet, and the animal walks on it, the chemicals can be absorbed into the animals bloodstream and cause neurological damage. I do not think this would cause bleeding from the nails, but one never knows, depending on the source of the chemical.
Let us know what you find out!!!!!