One popular formulation of Scientific Realism is: Mature scientific theories are (approximately) true.
One of the two main arguments against this claim is the so-called 'Pessimistic (Meta-)induction,' which is a very simple inductive argument from the history of science: Most (or even almost all) previously accepted, (apparently) mature scientific theories have been false -- even ones that were very predictively successful. Ptolemy's theory yielded very good predictions, but I think most people would shy away from saying 'It is approximately true that the Sun, other planets, and stars all rotate around a completely stationary Earth. So, since most previous scientific theories over the past centuries turned out to be false, our current theories will also probably turn out to be false. (There are many more niceties which I won't delve into here; an updated and sophisticated version of this kind of argument has been run by P. Kyle Stanford over the last few years.)
The kind of anti-realism suggested by the above argument is that the fundamental laws and claims of our scientific theories are (probably) false. But we could conceivably read the history of science differently. Many fundamental or revolutionary changes in science generate what Kuhn calls 'incommensurability': the fundamental worldview of the new theory is in some sense incompatible with that of the older theory -- the changes from the classical theories of space, time, and matter to the relativistic and quantum theories are supposed to be examples of such changes. Communication breaks down (in places) across the two worldviews, so that each side cannot fully understand what the other claims.
Cases of incommensurability (if any exist) could result in each side thinking the other is speaking incomprehensibly(or something like it), not merely that what the other side is saying is false in an ordinary, everyday way. An example from the transition from Newtonian to (special) relativistic mechanics may illustrate this: Suppose a Newtonian says 'The absolute velocity of particle p is 100 meters per second.' The relativistic physicist would (if she is a Strawsonian instead of a Russellian about definite descriptions) say such a sentence is neither true nor false -- because there is no such thing as absolute velocity. [A Russellian would judge it false.] If she merely said "That's false," the Newtonian physicist would (presumably) interpret that comment as 'Oh, she thinks p has some other absolute velocity besides 100 m/s; perhaps I should go back and re-measure.' To put the point in philosophical jargon: presuppositions differ between pre- and post-revolutionary science, and so the later science will view some claims of the earlier group as exhibiting presupposition failure, and therefore as lacking a truth-value, like the absolute velocity claim above. (Def.: A presupposes B = If B is false, then A is neither true nor false)
This leads us to a different kind of pessimistic induction: (many of) the fundamental theoretical claims of our current sciences probably lack a truth-value altogether, since past theories (such as Newtonian mechanics) have that feature. (If you want to call claims lacking truth-values 'meaningless,' feel free, but it is not necessary.) This is hard-core instrumentalism, a very unpopular view today; most modern anti-realists, following van Fraassen, think that all our scientific discourse is truth-valued (though we should be agnostic about the truth-value of claims about unobservable entities and processes). But this instrumentalism seems to be a natural outcome of (1) taking the graveyard of historical scientific theories seriously, (2) believing there is something like Kuhnian incommensurability, and (3) holding that incommensurability involves presupposition failure. And none of those three strike me as crazy.
Disclaimer: This argument has probably been suggested before, but I cannot recall seeing it anywhere.