I believe this is a question at the frontier of knowledge, i.e. basically a research question. From a quick google search, it appears to be mostly open. Also, for many years I have believed it to be important and linked to complexity theory lower bounds. You don't mention directly a statistical analysis but that is what is implied by your question. Here are two examples of statistical studies on DFAs/NFAs that are similar to show the general approach to questions of this type. It appears that basic empirical research into questions like this is still mostly unexplored. Admittedly the 2nd does not relate directly to your question but it's the closest I could find of current research.
To study your question, a statistical attack such as the following could be envisioned. Random NFAs are constructed. Then the minimal DFA is determined. Graph the histogram results of how many DFAs of size $x$ result. Separate out the "large" DFAs based on some threshold. Formulate some metric or measurement of the NFA that gives an estimate of the resulting DFA size.
This metric would be related to graph theory metrics such as edge density etcetera. There is probably some very important graph theory metric or mix of metrics that estimates the "blow-up" but its not immediately obvious to me. I could suggest something like graph coloring metrics or clique metrics maybe. Then test the metric against the two sets "blow-up" vs "not blown-up".
Other answers to your question so far only give an example case of a "blow-up" (useful for a case study) but do not address the key issue of a general metric.
Another area to look at a successfully developed program of empirical research is SAT transition point research. That has developed very deep links to physics and thermodynamics concepts. It seems likely to me that similar concepts are applicable here. For example, one is likely to find analogous transition point type metrics; probably edge density etc. Note parallels to Kolmogorov compressibility theory.
I conjecture also that NFAs that "blow-up" vs those that don't are quite analogous somehow to "hard" vs "easy" instances of NP-complete problems.
Yet another way to study this problem would be to formulate an NFA minimization problem. That is, given a DFA, find the minimal NFA, which last I heard (many years ago) was still an open problem.
 On the performance of automata minimization algorithms Marco Almeida, Nelma Moreira, Rogério Reis
 Automata Recognizing No Words: A Statistical Approach Cristian S. Calude, Cezar Câmpeanu, Monica Dumitrescu