Otoh, one of the early tests for general relativity was a minor deviation in Mercury's orbit from what was predicted by classical mechanics. My rough intuition is that inconsistencies can show up initially as faint flickers, but once you investigate more closely, one can find interesting and prominent counter examples.
I'm not an expert on this, but I believe the deviation in Mercury's orbit was easily observed and calculated, and thus became in a sense a binary call-out of Newtonian physics. Until highly precise measurements of relativistic velocities is common place, experiments like this are of course valuable and interesting, but one instance of an experimental result like this cannot be relied upon to supersede special relativity.
