Princess Alexandrine of Prussia, daughter of Prince Wilhelm, was born in 1915 with Down syndrome. Not much is written about her life, but she was featured predominantly in family photographs and was educated at the special education facility Trüpersche Sonderschule that was founded by Johannes Trüper.

Source: Horst-Heinz Richter: Johannes Trüper and his Sophienhoehe in Jena . Bussert and Stadeler, Quedlinburg / Jena 2003

Edit: Photo of the Prussian Royal Family http://imgur.com/wVKarMT

I can't speak as a historian but I can speak as a biologist. Mods, if this answer is inappropriate then please feel free to let me know.

Down's syndrome is most commonly caused by trisomy 21, which occurs as a result of cells failing to properly segregate their chromosomes during cell division. This means that disomy 21, the genetic change in the mother's egg that will eventually cause Down's syndrome, is usually acquired by maternal eggs rather than inherited. Trisomy 21 has a strong association with maternal age: the older the mother of a child, the more likely that child will be to have trisomy 21 or another non-disjunction effect. Given that noblewomen tended to marry and have children at a younger age, combined with the significantly increased risk of prenatal death that a nondisjunction disorder carries, the incidence of Trisomy 21 in noble families would be very rare indeed.

However, other disorders such as hemophilia in the Romanov family and mandibular prognathism in the case of the Hapsburg family likely had higher incidence rates in noble families than in the general population because of reproductive isolation. These disorders are hereditary rather than acquired and factors such as maternal age would not affect transmission. Furthermore, since many of these disorders are recessive then carriers would most likely be unaware that they are passing on the gene to their offspring, and so socially stigmatizing factors would not affect the gene's propagation.