Many authors have complained about the ill-conditioning associated the numerical solution of partial differential equations (PDEs) and integral equations (IEs) using as the continuously differentiable Gaussian and multiquadric continuously differentiable (C^¥) radial basis functions (RBFs).  Unlike finite elements, finite difference, or finite volume methods that lave compact local support that give rise to sparse equations, the C^¥ -RBFs with simple collocation methods give rise to full, asymmetric systems of equations. Since C^¥ RBFs have adjustable constant or variable shape parameters, the resulting systems of equations that are solve on single or double precision computers can suffer from “ill-conditioning”.’   Condition numbers can be either the absolute or relative condition number, but in the context of linear equations, the absolute condition number will be understood.  Results will be presented that demonstrates the combination of Block Gaussian elimination, arbitrary arithmetic precision, and iterative refinement can give remarkably accurate numerical salutations to even a (1000´1000) system of Hilbert equations.