My point about Mercury was important because it shows that there is an appreciable level of "give" that a theory has before the scientific community agrees that there is something wrong with it. That level of give is socially determined. It does matter if the discoverer of the anomaly is a Cambridge phd or a crackpot with no credentials. The measurement instruments matter, and the fallability of those instruments play into the acceptance of the results, too. A Popperian viewpoint is somewhat naive because what constitutes a falsification is incredibly fraught! Read Lakatos. He models scientific progression as a series of research programs that have "hard cores" of belief that are protected by ancillary theories. In the event of a negative result, it's those theories that are investigated first. For example, is my telescope correct? Is the theory of light that informs my telescope correct? Is there a dark planet influencing things that I can't see? In hindsight, Mercury should have falsified Newton, because all of the falsifying observations were valid. But it didn't because reasons.
We (and by we I mean Popperians) want to believe that science is a series of universally positive logical assertions that can be cut down by a single negative observation, as logic would dictate. But we don't always know what the criteria are for successful negative observations. The criteria are less rigid and well defined than we would be willing to admit. They vary from community to community. Robert Milikan won the Nobel prize for measuring the charge of an electron with his brilliant oil drop experiment. Only problem? His measurement was wrong. As folks tried to repeat it, they deviated more and more from his original measurement, until many repetitions and many publications later they landed on the correct value. If you were to plot the "true" measurement for the charge of the electron against time, you would see something deviating very slowly from an arbitrary incorrect value to the correct one. You have to ask, how on earth is this possible? Bias, authority, imprecision over truth criteria—all at play. And I think it's this sociological fuzziness in play in many thousands of small ways that lead us to at least question the assumptions on which truth is founded.