As noted on this Wikipedia page Newton's laws of motion and of gravity were "verified by experiment and observation for over 200 years" and found to be a "good approximation for macroscopic objects under everyday conditions". Now, of course, we know from Einstein and quantum mechanics that while not incorrect, his laws "only" work for a certain range of inputs.
While some of those early experiments were probably carried out with the hope of proving Newton wrong, nobody knew that there were limits to their applicability (very small scales; very high speeds; very strong gravitational fields) until much later (and, as far as I know, no one even suspected that there might be such limits).
As noted in the answers to this HofS&M question, there were some known "anomalies", such as the motion of the moon or Mercury, but these were more to do with the problem of applying Newton's laws to a three-body problem than problems with the laws themselves.
The question is: if Newton, or scientists around his time somehow had an "inkling" that there actually were limits to his laws, and set off on a "directed search" to explore those limits, would they have had the technology to expose such limits? In other words, would they have been able to explore a small-enough scale or move large-enough masses, at fast-enough speeds, and measure things with enough precision to detect any discrepancies above experimental noise and be reasonably confident in concluding that Newton's laws don't hold under all circumstances?