No, brain cells grow until you're 20 (when your brain is fully mature) then you only lose cells. I think its close to 40 000 per year after that, and you still have lots left at the end of your life, i wouldn't doubt that every little trauma causes the cells to die
EDIT: Not 100% sure on those facts, please correct me
This is not exactly right, either. The number of neurons (there are more kinds of "brain cells" than neurons, but the neurons are the ones that do the "thinking") actually peaks very early, because brain development produces a large number of redundant or unnecessary neurons. When an organism's brain receives various forms of stimulation, both from other parts of the body and from the environment via the senses, this reinforces certain pathways over others, and neurons that don't get "used" are programmed to die. So then early in life, the number of neurons peaks and then decreases due to this pruning, according to plan.
As an example of how the number of neurons doesn't really exactly correlate with an organism's intelligence (really, the quality of the connections is more important), the cerebellum, a relatively small structure in the rear of the brain, which is very important for stitching thoughts and movements and things together and making them smooth, but which doesn't primarily handle your reasoning, speech / language, or other aspects of your "knowledge," or store memories, or anything like that, likely has something around 85% of the neurons in your brain, with the other 15% making up the cortex (where the thinking, perception, etc, happens) and all the other deep structures too.
Anyway, the number of brain cells peaks early, and then pruning occurs. After that, the development isn't in the growth of new neurons so much as it is in development of connections
between neurons. Each neuron has two kinds of connections, in essence -- dendrites, which are sort of like its inputs, and an axon, which is sort of like its output. The dendritic development happens more or less continuously throughout life, to some extent. The axons of some major neurons develop an insulating layer, called myelin (which makes them transmit better, sort of like an insulated wire works better than bare metal). When you said that development continues into one's 20's, usually
that -- myelinization of major pathways -- is what people are talking about. Most of that happens from late childhood / early adolescence to the mid to late 20s and maybe a little into the 30s.
Meanwhile, the brain
does actually continue to build new neurons in certain regions, but we're honestly not entirely sure why or what the new neurons are used for. In late life, when the brain atrophies, part of this is some cell death and part of it is also just a loss of the richness of connections.