I was recently trying to convince someone that that Special Relativity (SR) was more correct (in the situations where it is applicable) than classical (Newtonian) physics. One argument I used is that irrespective of experimental evidence for SR, SR is actually a simpler theory than classical physics, when you write each theory down in their simplest forms.
In this form, physics looks very different from classical "high school" physics. A lot of concepts which are classically very different turn out to be the same thing in relativity. Space turns out to be the same thing as time. Different angles (as in rotation) turn out to be the same thing as motion at different (constant) velocities. Electric fields turn out to be the same thing as magnetic fields.
This theory has a "parameter", a value which isn't predicted by the theory and must be determined experimentally and plugged into the theory to make it complete. This parameter is called "c" and is usually known by its physical meaning "the speed of light in a vacuum".
Now, one could conceivably get classical physics in this same form by plugging in a "c" value of infinity instead of 299,792,458 metres per second. Doing this causes time and space (and angles and velocities, and electricity and magnetism) to separate out like an emulsion of oil and vinegar left to stand for a while. Only the finite value of c causes these concepts to mix (and the smaller the value of c, the more they mix and the more pronounced relativistic effects become).
"Great," you might say, "so Einstein might have been wrong all along and all this weird time dilation/length contraction/mass equals energy stuff could all be bunk." The trouble with that, though, is that with c=infinity, the model corresponds less well to observed experimental results - the time dilation effects that have been measured are not predicted, and the speed of light is predicted to be infinity.
But the place where this model diverges most drastically from reality is magnetism. The c=infinity theory predicts that there should be no magnetism at all. Trying to add magnetism back in to a non-relativistic theory causes all sorts of complexities and irregularities. In fact, it was trying to remove these irregularities that brought about relativity in the first place. Really, the simplest way to have a consistent theory of magnetism is relativity with all the non-intuitive concepts that entails.