The brain does generate new neurons over a lifetime, a process known as neurogenesis, but at a sedate pace. Various methods have been shown to boost the rate of neurogenesis in laboratory animals, in order to somewhat reverse age-related loss of memory or turn back some of the other symptoms of neurodegenerative conditions. It isn't always clear that the generation of new neurons is in fact the direct cause of such benefits, however:
[Researchers] have discovered that by reestablishing a population of new cells in the part of the brain associated with behavior, some symptoms of Alzheimer's disease significantly decreased or were reversed altogether. The research [was] conducted on mouse models; it provides a promising target for Alzheimer's symptoms in human beings as well. "Until 15 years ago, the common belief was that you were born with a finite number of neurons. You would lose them as you aged or as the result of injury or disease. We now know that stem cells can be used to regenerate areas of the brain."
After introducing stem cells in brain tissue in the laboratory and seeing promising results, [researchers] leveraged the study to mice with Alzheimer's disease-like symptoms. The gene (Wnt3a) was introduced in the part of the mouse brain that controls behavior, specifically fear and anxiety, in the hope that it would contribute to the [expression] of genes that produce new brain cells. Untreated Alzheimer's mice would run heedlessly into an unfamiliar and dangerous area of their habitats instead of assessing potential threats, as healthy mice do. Once treated with the gene that increased new neuron population, however, the mice reverted to assessing their new surroundings first, as usual.
"Normal mice will recognize the danger and avoid it. Mice with the disease, just like human patients, lose their sense of space and reality. We first succeeded in showing that new neuronal cells were produced in the areas injected with the gene. Then we succeeded in showing diminished symptoms as a result of this neuron repopulation."