There have been a number of experiments using super-accurate clocks to measure time dilation. In one case they flew them on planes and showed they disagreed after landing. In another they put one atop a water tower and one at the base (light moves slower the deeper in a gravity well thus time moves differently).
I suppose it is a nod to the accuracy of these clocks that they can discern this difference which, in these cases, is quite small.
But as long as I’m here, I’d like to add that we trust these clocks because Quantum Mechanics has held up remarkably well as a theory; I’m tempted to state it as a fact, but I’m wishy washy.