So off and on I've been studying basic recursion theory and I've realized that, at least when restricted to the basic stuff I've been learning, recursion theory is essentially the study of uses of diagonalization to get non-recursive sets / classes. So there's a lot of existence proofs. 

Are there "limits" to our applications of diagonalization? In other words, is there a point at which our diagonalizations fail to carry us from one class to another?

Thanks!