Can good theories be falsified?

See also The Revolution That Didn't Happen (by Steven Weinberg)
Weinberg's essay explains the limitations of Kuhn's theory of scientific revolutions in similar way as the present essay explains the limitations of Popper's theory of conjectures and refutations. .

The philosopher Karl Popper claimed that falsifiability is the hallmark of scientific theories. But scientific practice speaks against him.
A correct theory cannot be falsified, and in this sense is not falsifiable, in spite of Popper. (Falsifiability can be asserted only in a contrafactual sense, that there are _conceivable_ situations that, according to the theory, are excluded. But for a correct theory, these situation will never happen, hence are completely ficticious.)

What happens with good theories is, at worst, that their region of validity or accuracy gets restricted as new data about more remote instances come in.
In today's understanding, people are careful to indicate the limits where a theory is claimed to be valid, and the accuracy to which its answers are to be trusted.
For example, the Standard Model is claimed to be valid whenever gravitation is negligible, accuracies conform to present possibilities, and energies are well below a putative unification scale. Failures outside this domain are not counted as falsifications.
While limits and accuracy claims are not necessarily part of the theory proper, they are part of the theory as actually taught and applied. Indeed, although people try to extrapolate, one can never be sure whether a theory is correct outside the domain where the data were collected.
But one can be reasonably sure within the domain where enough data are available. Good scientific practice requires that a good theory agrees with the data within the tolerances claimed. Once this is the case, these theories can never be falsified. Rather, if people find disagreement in experiments, the theory falsifies the experimental arrangement or analysis.
All science students who ever did experiments in the lab know very well that this is common practice.

The degree of caution and care at the highest level of quality has been increasing through the centuries. It is now too late to ask Newton whether he believed his theory was valid without restrictions. (Or are there any hints in the Principia Mathematica?) Certainly Newton's theory as taught today is valid as taught (i.e., with the restriction that speeds are small compared to c and at distances large compared to the radius of the largest atom). But we nevertheless believe that it is the 'same' theory, and if Newton would live today, I think he would agree with that.
And Newton's theory will never be falsified, unless God suddenly decides to change the physics of the Universe.
(That the observed advance of Mercury's perihelion did not match Newton's theory was known as a limitating condition already before relativity was born.)

The meaning of ''Newton's theory'' is not invariant under time. Newton might not even recognize ''Newton's theory'' as taught today as his theory. The conceptual basis has shifted in the many years since his time. Nobody knows whether Newton regarded his theory as something universally valid for arbitrarily accurate observations. It seems however likely to me that he never claimed this.
The point I made above is that to the extend a theory has been found reliable at some point in time, it is reliable to the same degree at all other times (unless the laws of Nature change), irrespective of any falsification outside this core region.

Falsification of a theory correct in some domain in a previously untested domain commonly leads to an extension of the theory, not to its downfall. Otherwise Newton's theory would have had the fate of Phlogiston, which died as a theory long ago as it was found to be a truly misleading theory.
Real advances in theory - such as Newton's progress over Ptolemy and Kepler, or Einstein's progress over Newton - usually lead to simplification that unify many observations. This is one of the hallmarks of theoretical progress.
But even Newton's simplifications leave the Navier-Stokes equations (derived much under some assumptions from Newton's theory, extended to the microscopic world) an unresolved mathematical region that needs uncontrolled approximation heuristics to get high quality results. Such heuristics allows checking (hence verification or falsification) on a heuristic level only, as one never knows for sure (i.e., in a mathematically rigorous way) how far off the true results are.
Quantum physics also has much mathematical simplicity, which makes it an ideal tool to unify far more observations than Newton's or Einstein's theory did.
However, the subarea of quantum physics called relativistic quantum field theory (RQFT), treating the subatomic world of elementary particles, is currently still in a state not so far from pre-Kepler astronomy. The Feynman diagram expansion in RQFT are not very unlike the epicycles of Ptolemy, to be replaces by better ''nonperturbative'' methods once they will be found. Surely, here some simplifying progress can be expected, though the problem seems to be so hard that there is no predictable road towards success.