Renormalization is always needed when the Hamiltonian is singular. Singular means that the formal expression for the Hamiltonian resulting from the interaction specified is not a self-adjoint operator in a dense domain. Then the dynamics is formally ill-defined and must be renormalized by taking care to represent everything properly as a limit that makes sense.
In particular, this is always the case in interacting relativistic quantum field theories in 3 or 4 space-time dimensions.
To understand why and how renormalization works one can first consider simpler situations in quantum mechanics. In this case there are explicitly solvable toy models (low rank singular perturbations of simple solvable systems) where one can see exactly what happens and why. See my paper Renormalization without infinities - a tutorial, which discusses this in detail.
About how (or whether) one can tell whether the terms in an asymptotic series will be small see the discussions in Chapter B5: Divergences and renormalization of my theoretical physics FAQ.
