Attempted and entirely hypothetical example from astronomy. (There may be problems with this example; I could be misinformed or have misconstrued any of a dozen things about astronomy & physics).
Joe Astrophysicist finds that equations for the distribution of mass, assuming rapid expansion in the moments immediately after the convergence-point of singularity, could call for really massive torus-like rings that would be emphatic in the outer expanding layers, then smooth out and gradually disappear in successive inner layers as initial expansion continued. There is no sign of such massive concentrations up to the limits of what we can perceive (~ 12-15 billion light years away from us, beyond which the transit-time of light would be greater than the age of the universe). But if there were a great deal of mass out there in the range of 20-30 billion light years distant, that would explain to a significant degree the apparent acceleration of the rate of expansion observed in the red shift of far-distant objects: they are being dragged along by objects yet more distant, the existence of which is something we cannot test for and will never be able to test for (at least not directly) because any EMR emitted by anything that far away will not have had time to reach us.
OK, pause for laughter at my fledgling grasp of astrophysics, but that type of thing represents a legitimate form of theory which helps explain observable phenomena, yet which posits, as part of its explanation, the existence of things we cannot verify with empirical testing. In fact not only can we not do so now, but are intrinsically not going to be able to do so (or at least not any time in the next 8 billion years or so).
If you want something that more closely resembles actual theories and not hypothetical ones I’ve invented for the sake of argument, how about string theory? I could not begin to explain string theory to you or anyone else, but I gather that the following are true:
• It is appealing because it explains certain phenomena
• It has yet to yield up a single falsifiable hypothesis
Feel free to correct me if I’m wrong.
Or, hey, I seem to have just stumbled over a dead horse that someone’s been beating on not too long ago. Evolution, specific fossil specimens in the fossil record, a theory that extrapolates from one set of fossil records (one ‘dot’ if you will) to another, postulating that the former evolved into the latter via natural variation and natural selection that could hypothetically be observed originating in individuals and then being passed on to proliferate, if we could see all of the individiuals of the species and fast-forward… as many a Bible-thumping creationist has asserted, we don’t get to see that because we don’t have that nice Mighty Morphing Evolution Rangers movie to watch and instead only have some scattered data points and some extrapolation, plus some other corroborating information here and there. It is, in my opinion, a very compelling extrapolation. (I don’t see how anyone could look at the progression from Eohippus to modern Equus and walk away not believing in evolution). But it really does assume the existence of data (species, specimens, specifiable if not as of yet specified changes in specific parts of DNA that we have no samples of, etc) that we believe is real and yet for which we have no direct empirical data. We believe those things because their existence is called for by the theoretical model by which we made sense of the data we do have.
(You do believe in the real existence of the mutations in the Eohippus DNA that we don’t have on hand that represented steps towards becoming Merychippus and so forth, yes?)