The inverse and saturation theorems for radial basis function interpolation are investigated. It is shown especially that a function that can be approximated sufficiently fast must belong to the native space of the basis functions in use. The main result in case of inverse theorems deal with basis functions that generate Sobolev spaces as their native spaces of a certain smoothness class. Certain saturation theorems are elaborated for the case of thin plate spline interpolation. It is shown that functions that can be approximated with a high-order are necessarily polyharmonic functions. The new characterization of the native space generated by radial basis function interpolants is provided. It gives a numerical tool to test whether an unknown function belongs to the native space (and thus in several cases to a Sobolev space) or not.