Some problems are just damn hard. I don't know enough Physics, but in CS there are many examples of problems that have been open for decades were we still don't even know where to begin with a proof. The most prominent example is P vs NP. It's been open for forty years and all we have is a number of proof techniques where we know that they can't work. Complexity theory in general is a field where all the interesting problems either are solved or have been open for a long time and researchers retreat to working on very special cases of very special problems.

Figuring out the null hypothesis (toy model) can be a big first step. For instance, planets should move according to an ellipse, but they don't because of the gravitational pull of other planets and moons. This deviation from the null hypothesis can be quite informative and help with further discovery.

Well, physicists know that some of the premises are fundamentally wrong. For example, the strong suspicion is that space (or time) it not continuous and smooth at it's deepest level. But finding a solution is a whole different thing.