Short answer: Yes, we do shrink with age.
The most important reason is that the cartilage in the joints between our bones gets worn out and thinner, as well as disks between the vertebrae of the spine. This results in a compression of the spine and also to a loss in height. Shrinking bones due to osteoporosis can also play a role, as well as muscular atrophy due to ageing.
According to "The Baltimore Longitudinal Study of Aging" (reference 1) adults start shrinking by the age of 30 by 5-8mm per decade. This is relatively short after you completely stop growing. This process gets faster with increasing age, the study says it goes faster after 70.
Men and women do not shrink equally, men loose around 3cm between 30-70 and a total of 5cm by the age of 80. Women loose about 5cm between 30 and 70 and a total of 4.5cm by the age of 80 (these are average numbers from reference 1). Reference 2 also has a nice summary of this original article.
This effect can be attenuated by regulary excercising as the study listed in reference 3 shows.
- Longitudinal Change in Height of Men and Women: Implications for
Interpretation of the Body Mass Index: The Baltimore Longitudinal
Study of Aging
- Yes, You Are Getting Shorter
- Role of physical activity training in attenuation of height loss