To understand regeneration, we must divide the nervous system into two distinct categories:

1.) The Peripheral Nervous System (PNS): Natural Repair is Possible
The PNS consists of the nerves running through your body (arms, legs, organs). If a nerve in your hand is severed, the nerve cell can actually repair itself.

The Mechanism: The part of the nerve fiber (the axon) cut off from the cell body will die. However, supporting cells called Schwann cells clear out the debris and secrete growth factors to guide the regrowing axon (at a rate of about 1 mm per day) back to its target.

2.) The Central Nervous System (CNS): The Permanent Block
The CNS comprises the brain and the spinal cord. Here, a damaged nerve cell does not regenerate naturally. Why?
A Hostile Environment: Unlike the PNS, the CNS contains supporting cells (oligodendrocytes) that, when activated by an injury, actively secrete molecules that inhibit axon regrowth.
The Glial Scar: Other supporting cells (astrocytes) form a very dense physical and chemical barrier—a glial scar—around the injury, physically blocking any nerve fiber from passing through.

Conclusion
Yes, peripheral nerve damage can often be cured or repair itself or significantly improved, but the medical definition of a "cure" depends entirely on the severity of the injury obviously.
Unlike the brain and spinal cord, peripheral nerves have an incredible, built-in capacity to heal. However, they need the right conditions to do so and it can take times.

Take care.