The Ultraviolet Catastrophe was discovered immediately after theoretical physicists derived the radiation law, the limit was theory and not experiments there as well. And the most important part here is that another physicist had already derived an alternative radiation law at the time that fit perfectly with the observed deviation and had a good explanation for it: Quantum Physics.
Edit: The problem with physics today is exactly like back then, we have no predictions to test. If someone comes up with a new theory that joins quantum physics with gravity in a way that is consistent with all past experiments, then we can test that. But there is no such theory today, nobody has figured out a way that the domains can work together.
In https://news.ycombinator.com/item?id=29144119 I listed a lot of the big unsolved problems in physics, though many would argue that the question of how consciousness arises isn't part of physics. In https://news.ycombinator.com/item?id=29153894 I also listed a lot of recent advances, many of which came from experimental physics. Some of those problems are susceptible to experimental investigation even with the theories we have.
But, aside from these big problems, many smaller problems are susceptible to experimental investigation. You may not create an Einstein-style paradigm shift by detecting CNO-fusion neutrinos from the sun, observing Higgs decay, improving quantum-dot solar cells, fabricating nanotube rope whose strength approaches that of a single nanotube, understanding the lubricity of BAM well enough to design more similar materials, deriving useful energy from the fission of lithium, making a metamaterial with a higher Young's modulus than diamond with negative-elasticity inclusions, constructing logic gates out of fluid vortices, building a usable hypersonic plasmoid pistol (one that doesn't require explosive flux compression pumping!), building transistors that function at 800 degrees, finding a SHS route to cubic boron nitride, making a 50-tesla magnetic field in the lab, finding a way to construct quantum-dot solar panels that's cheap and scalable enough to undercut poly-Si, confining a particle of gold in a stable minimum of the Casimir potential, or finding a way to fabricate high-quality optics apparatus rapidly out of aluminum foil; but the obstacles to these problems are (or were) mostly not that we don't have any useful hypotheses to test.
Edit: The problem with physics today is exactly like back then, we have no predictions to test. If someone comes up with a new theory that joins quantum physics with gravity in a way that is consistent with all past experiments, then we can test that. But there is no such theory today, nobody has figured out a way that the domains can work together.