Great question so I took a quick look.
In 1985 there were three. Villanova, Georgetown and St. Johns. Memphis was the fourth school.
I can't find one with four.
With schools like Kansas, Kentucky, North Carolina , Ohio St.,Michigan St. and UCLA being in it so much it becomes kind of hard to get 4 privates.
There have been 82 Final Fours before this year. In 63 of them at least one of the above mentioned schools has made it.