This paper deals with the relationships between the stability bounds of a boundary value problem (a first-order linear system with \(B_ 0y(0)+B_ 1y(1)=b;\) \(B_ 0,B_ 1\in {\mathbb{R}}^{n\times n}\), \(b\in {\mathbb{R}}^ n)\) and the growth behaviour of its fundamental solution. Introducing certain fundamental concepts of dichotomy, dichotomy bounds from bounds for the Green function are given. This is extended to the case of exponential dichotomy. The reverse result, viz., that the dichotomy implies well- conditioning, is presented. Finally a number of numerical examples to demonstrate these concepts is given.