High order accurate inviscid flux discretization schemes have been used for many years in the context of block structured CFD solvers. In order to address complex and moving geometries, many of these solvers incorporate overset composite grid techniques. The vast majority employ variants of Lagrangian interpolation to determine overset donor weights and, of these, most use 2nd order accurate interpolation stencils. This paper demonstrates the pitfalls of using lower-order (i.e. 2nd) overset interpolation strategies in conjunction with high-order solver numerics. Simple theory and familiar canonical problems are used to demonstrate that in certain types of problems failure to use high-order interpolation can quickly lead to errors that undermine the reasons for using an advanced numerical scheme in first place, namely accuracy. Results also include calculations of a complex geometry with overset grids in relative motion, and their comparison to test data. Cases using standard and high order overset interpolation show the value of advanced interpolation schemes. Two computer codes are used: OVERFLOW 2.2, and a high order code developed at Penn State.