The classical Rado-Horn theorem describes the smallest number, R, for a given family of vectors such that the family can be partitioned into R linearly independent subsets. In joint work with Bodmann, Casazza, and Speegle, we have obtained some refinements of this theory in cases where one has additional information on the set of vectors. For example, when the set of vectors is a uniform Parseval frame. In this talk we will introduce the classical Rado-Horn theory and present two of our theorems.