Yes, people do die from the neurodegenerative disorders that cause dementia, such as Alzheimer's disease (AD). The last time I looked for official statistics (to answer a question about the youngest age at which someone had died from AD), there were 65,965 deaths from AD in the US in 2004. (And the youngest person to die from AD that year was in the 15-24 years old age range.)
Deaths from disorders such as AD are significantly under-reported, because the source of information for these statistics (which are compiled by the US Centers for Disease Control and Prevention, National Center for Health Statistics) is typically a death certificate, and a death certificate often lists the immediate cause of death (e.g., dehydration, malnutrition, bedsores, urinary tract infection, or pneumonia, all of which are commonly associated with death due to a dementia) as the primary cause, with AD sometimes given as a contributing cause, if listed at all. The Alzheimer's Association is working to change this practice, since it is very misleading.
The report, 2011 Alzheimer's Disease Facts and Figures, indicates that AD is now the sixth-leading cause of death in the U.S. (And that, of course, does not include deaths due to the other dementias.)