Yes - Iowa had one in Glenwood in Mills County on the far western side of the state. You can read about it by clicking on the following link:http://en.wikipedia.org/wiki/Iowa_Institution_for_Feeble-Min...
While it was originally established as a state funded Civil War Orphan's home, in a few short years it became the Iowa State Asylum for Feeble Minded Children. It had a pretty horrible history and became a place where people dumped their children, either because they couldn't handle them or couldn't afford them.
I found that my Great-Grandmother had been listed there in the 1880 Census as an "inmate". 9 years later she was married to my Great-Granddad. I called the institution as it still exists today under a far more socially acceptable name. They have records from the 1800's and I called to find out who put my G Grandma there. They claimed she was never there until I provided photos of the actual census documents. They relented that some of the children on the page they had records for and some not - claiming they must have been remnants from when the institution was an orphanage for Civil War Soldier's Children. When I pointed out the Civil War had ended 8 years before her birth, I got a shrug. All of the family records show that she was a sweet, kind and hard working woman who raised three normal children and cared for a successful husband. Nothing indicates she was developmentally disabled in any way, so I suspect she was a victim of harsh financial times for her parents post war and she was "dumped" there.
Thus is the history of this fine institution!