Quantum field theory defines its own bounds of applicability I recall hearing in a lecture something along the following lines: 
"Due to some intrinsic feature of quantum field theory in general (or maybe it was the standard model?), we know where it is applicable, and thus we know that there will be no new physics at low energy (the part I care about is in day-to-day living), or at least not without huge (like throwing away relativity) changes in our models."
What is this property at a formal level?
Is there a concise way to explain it to non-specialists?
 A: My guess is that this statement refers to the concept of effective field theory, which is part of the quantum field theory framework, and also present in the standard model. This approach was invented by and is named after Ken Wilson ("Wilsonian effective field theory").   
The idea is to formulate a theory that is not valid up to arbitrarily high energies, but only up to some cutoff. In this sense, such low energy theories are just approximations to a full theory that is assumed to hold up to the Planck scale. The structure of the theory is dominated by this cutoff, and if you translate the latter as "bound of applicability", the answer to your question should be evident. 
To be more precise, when one constructs an effective field theory, important parameters like coupling constants and masses will in general depend on this cutoff. The requirement that physical observables are independent of it determines the precise dependence. This is realized mathematically by the renormalization group equations. In this context, the term "Wilsonian renormalization group" is used. 
An important advantage of this is concept is that it allows a priori non-renormalizable theories to make sense. One should also note that the standard model is also thought of as an effective field theory: it is assumed that it only holds up to a certain energy scale beyond which we will deal with new phenomena (e.g. supersymmetry, string effects).
This is as much as I can say without getting into technicalities. If you are intererested in more detail, I can recommend for example chapter 29 of Srednicki's book on quantum field theory. 
