Yes, brain cells are lost as we age. In fact, brain cell loss begins in the 20's. In addition to the loss of brain cells, the brain makes less chemicals that it needs to perform work. This has an affect of both the short term memory and the long term memory. Keep in mind that this is a very gradual loss. We do not lose enough brain cells to be forgetting things left and right. We have many brain cells so the few that are gradually lost are not enough to cause any kind of substantial memory loss. There are many other things going on in the brain. When memory starts to fade it is usually not due to the loss of brain cells.
The short answer is yes. But you don't have to wait until you get older, it's actually happening right now, and has been since you were 18 or so. Brain cells do not regenerate, and we lose them at a slow rate over the course of our entire adult life. Not to worry, though, as you have billions of them, and as we age, it is usually other disorders and issues with our brains that cause us to lose function or memory rather than cell loss.
One of the reasons stem cell research has been so intriguing is that it is one o the few treatments we have found over time that may be able to reverse diseases of the brain such as Alzheimer's.