So here's my question. It seems like we want something like reproducibility when we develop and run scientific experiments, but there are acceptable exceptions to this rule. So what is the rule? The responses running around in my head are either far too weak or far too strong.
So. So far, in your post, we have two types of legitimate studies- reproducible studies, and theoretically reproducible studies. But, as you pointed out, the latter of the two is a somewhat nebulous concept. And, as I'm about to point out, so is the first.
In order for a thing to be objectively reproducible, it must be perfectly reproducible. But think about it. So you have a study where you do thing A, get results B, and make conclusions C. Someone wants to verify C, so they will try to replicate A in order to see if they get B or not. But A can never happen again, because part of A includes the time and place that A happened. A can at best be approximated.
And so really, the first category is not reproducible, but rather
sufficiently reproducible. Which is a very subjective term. Reproducibility is subject to, among other things:
general consensus of the scientific community
The strength and nature of the claims C that were made from data B
The context of A and B
How well A is explained
I think the input of the community is often downplayed, but it is a very important component. It comes into play largely via the scientific publishing system, which is based on peer reviewing and significant editorial control. Basically, it all comes down to what is and is not "okay" in terms of publishability. Often, things are "okay" until someone demonstrates why it is not okay, which doesn't exactly follow the ideals of the scientific method.
An example, in case I've been rambling too much:
Say someone does some study looking at the effect of diet on the common rat, rattus rattus. At the time, its considered okay to try to reproduce a study as long as you use the same species. So, one would NOT be reproducing the example if they used mice instead of rats.
But then, someone comes along discovers that rats from different parts of the country react differently to diets. Which means that in order to replicate the example, you would need to use rats from region that the original researcher got rats from.
But. It also depends on the claims that the researcher made- did he say that his results applied to all rats, or just the rats he used in the study?
And here's the kicker- if at the time the researcher did not think that regionality was important, he might not have even listed where his specimens came from. Which, given the new evidence, significantly changes the reproduciblity of the experiment.
TL;DR:
Reproducility is not an innate characteristic of an experiment, but rather subject to a variety of forces.