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Before Einstein came up with General Relativity, was there any serious reason to doubt Newton's theory (and its various developments)?
I only know about the discrepancy in Mercury's orbit, which may have been caused by an error (e.g. similar to the one that caused people to make up Planet X).
You raise an interesting point about the role of experiment and falsifiability in science. Despite a long-standing anomaly in Mercury's perihelion, Newton's theory of gravity itself wasn't heavily questioned, let alone rejected or falsified. Rather, auxiliary assumptions were concocted that saved Newton's theory, such as an erroneous mass of Venus, a planet inside Mercury's orbit and that Mercury had a moon.
Imre Lakatos developed this idea in his criticisms of falsifiability, see e.g.
Auxiliary hypotheses are considered expendable by the adherents of the research programme—they may be altered or abandoned as empirical discoveries require in order to 'protect' the 'hard core'. Lakatos was following Pierre Duhem's idea that one can always protect a cherished theory (or part of one) from hostile evidence by redirecting the criticism toward other theories or parts thereof (wiki)
Kuhn explicitly discussed the case of Mercury's perihelion in his seminal The Structure of Scientific Revolutions (1962):
No one seriously questioned Newtonian theory because of the long-recognized discrepancies between predictions from that theory and both the speed of sound and the motion of Mercury ... [The discrepancy] vanished with the general theory of relativity after a crisis that it had had no role in creating.
Einstein, for example, seems not to have anticipated that general relativity would account with precision for the well-known anomaly in the motion of Mercury’s perihelion, and he experienced a corresponding triumph when it did so
The development of general relativity followed from theoretical inconsistencies between Newton's theory and Einstein's relativity, rather than evidence that contradicted Newtonian theory.
That said, Newton's theory didn't have to wait for Einstein for theoretical criticisms. Even in Newton's day, his theory was criticized on conceptual grounds as being "occult," for it permitted action at a distance, especially by Leibniz. However, Kuhn suggests that these criticisms died away and only returned in light of general relativity:
When Newton’s theory had been accepted, a question [the origin of gravitational attraction] was therefore banished from science. That question, however, was one that general relativity may proudly claim to have solved.
Mach et al. criticized Newton's absolute space and formulated alternative explanations of Newton's bucket experiment.
More recently, Schrödinger applied Weber's force law to gravitation/cosmology:
cf.:
