The Higgs boson is actually an excellent example; it's one of the rare things in physics that cannot be proven to exist without first measuring it. This is actually quite interesting, as technically it should not be the case.
Not so rare. Frexample, to paraphrase something interesting I read not so long ago (better than an example I could make up out of thin air), scientists making a certain type of Carbon Dioxide laser when it was first developed were sometimes making ones that worked and sometimes making ones that weren't. But it was quite obvious when they made ones that worked because they'd make a beam that was detectable. And, by all accounts, capable of melting concrete... a pretty good sign that it's working.
OTOH, scientists trying to construct interferometers to detect purely theoretical gravity waves were sometimes getting an apparent signal, and sometimes not. The problem was that Gravity Waves might or might not exist (a bit like the Higgs Boson), depending on whether current theories (and, indeed, which of them) were correct. So were the non-working ones not detecting because they had been made wrongly (a problem previously encountered with the laser, even when directly copying the instructions written by the working-laser constructors), not looking at the right time/in the right direction, not detecting because there was nothing to detect (the working versions being built incorrectly in order to false-positive) or even that the interferometers that were detecting
something were detecting a different something to what they were purposefully intended to detect and the non-detecting ones were not detecting this something else because they were not tuned to this alternate phenomena... Among other possible answers to the question of why some apparently worked and some apparently didn't.
And recently, when I was at a talk with an expert in anti-matter[1], the question of whether some of the unknown questions regarding the CPT symmetry would be solved by string theory was responded to by an answer that boils down to "if (string_theory == true) then (its a whole new ball game)". OTOH, people have (different) expectations of what the more 'classical' theories should show and have (or are starting to have, thanks a relatively small offshoot of the CERN ring and some ingenuity) the means to test against these expectations and rule out the obvious wrong answers for any particular theory structure. Later on, they may be able to rule out all non-String Theory answers (or at least all current ones), or even come up with something that means that the (rather loose) String Theory concept needs revising to agree with experimentation. But String Theory is about as far into the "God Did It" territory of unprovable conjectures as most mainstream physicists will even consider treading[2]. To grossly simplify the matter, of course.
TL;DR: There are various levels of "good enough" explanation, and things like the Higgs can be imagined to sits in the cracks in the most finely smoothed explanations once we start looking at the "wall of science" close enough to spot the imperfections in its plaster. As might the God Of The Gaps, but that way lies Occom, glowering at one whilst brandishing his infamous shaving equipment.
[1] Oh boy, that does sound almost like intellectual name-dropping... But it is a relevant exemplar.
[2] Like if Priestly derived the electron/proton/neutron nature of atoms from his experiments with gasses in the 1770s... He would have been right, but without any reason to be. Rutherford discovered part of the answer through scattering, almost exactly 100 years ago, through means not available to Priestly (and Lavoisier and Scheele), and the neutron remained invisible to science until the 30s, IIRC. And if that's not 'actually how the universe is' (or, rather, not a sufficiently accurate simplification of the actual state of affairs, when stripping away all our anthropocentric abstractions and dealing with the raw universe) then we haven't built (or correctly assembled/applied) the tools required to prove that this is not the case.