Brain cells do regenerate but only to a certain extent.
Your brain cells regenerate through a process called neurogenesis.
Neurogenesis occurs in specific regions of your brain like the hippocampus and the olfactory bulb throughout your life.
The neurogenesis process is also facilitated by neural stem cells that reside inside the brain and have the potential to divide and differentiate into new neurons and other brain cells.
The brain can also repair itself to a certain extent depending in injury.
In response to a brain injury, the neural stem cells of the brain can become activated and produce new cells in order to replace any damaged ones.
This process of producing new stem cells is called regenerative neurogenesis.
However neurogenesis is limited in it's ability, scope and efficiency.
And the amount of new cells generated through neurogenesis is often small when compared to the total number of brain cells.
And any newly formed neurons may not always function perfectly or integrate fully into the existing neural network of the brain.
Age, genetics, exercise, diet and environmental factors also play a role in the ability of the brain cells to regenerate and repair itself.