And at what point in its development and study do you abandon a theory that nobody has yet invented a way to test?
Theories like String Theory tend to fall into the first, possibly dabbling with the second option. Researchers are already starting to work on ways to test string theory, with promising results. It's quite likely that within 20 years, we might be able to do just that. While things like the many worlds interpretation of quantum mechanics or multiverse theories tend to fall on the "maybe in 1,000 years, but probably never" category.
I think the real question is: How patient are we?
If you mean we're going to be hearing about the Many Mystical Worlds of Quantum Physics from Deepak Chopra types for the next 1,000 years, I'd say "not very" is a reasonable answer.
Perhaps a better question is at what point does philosophy become physics? Finding a way to test the theory, and doing so, may be such a point.
String theory's critics are sure that it isn't. I've even heard it said that it only has the prominence it has because a noisy generation of physicists elbowed out competing projects.
If you don't have a testable theory in the short term, should you carry on trying to refine that theory in the absence of evidence, or should you spend more time on competing theories that perhaps have more chance of being open to experimental verification in a reasonable time frame?
The list of open, unsolved, and unexplained issues in physics isn't small. Continuing to prioritise one theory and gambling that it works out eventually is a classic sunk-cost error.
https://en.wikipedia.org/wiki/Atomism
You could also derive the Atomic nature of matter through math though, as infinitely devisable sets end up with paradoxes that don't match up with reality (think banarch-tarski)
You can do all this without an idea how to test it.
The philosophers that invented atomism probably thought about how the might see their atoms directly. I wonder how many "crazy" ideas were dreamed up that sounded impossible, that are now easy experiments to do today.
Theory is important, even if it sounds impossible today, because we have a history of redefining what is "possible" throughout the history of science.
Clearly if you propose a theory that requires an experiment so advanced (or circumstances so rare) we can't hope to do it in 20 years, it's still a valid scientific theory. What about 50-100 years? What if it requires technology so advanced it's nearly unthinkable that we will ever attain it? This is when it becomes a philiosophical gray area. It's not a clear cut case what is verifiable and what isn't, since the theory is valid before our ability to verify it.
Theoretical physics is also very much a thing.
> and should therefore be testable by definition.
That sounds nice but the definition of testable is hard to pin down and changes over time.
What technology can you imagine that could ever prove or disprove a hypothesis of our universe being only one facet of a multiverse? The apparatus would necessarily exist within the closed system.
does being in a multiverse influences our world in any way using current theories?
yes > we can eventually figure out and run an experiment no > then studying it is as productive as searching the Russel's teapot
this is how you set apart a theory from speculation, a falsification test. i.e. > http://arxiv.org/abs/1210.1847