How little scientists know who only know science!  Thanks again to Norm, I learn about some statements by a retired professor of chemistry, Peter Atkins, about how we know what we know.   Atkins is quoted as saying:

The scientific method is the only reliable method of achieving knowledge.”

Well, first, it is worth saying that the scientific method does not produce reliable knowledge.  One of the two defining features of science is that scientific claims are defeasible:  they may be contested, questioned, challenged, and even overthrown, if the evidence warrants.   There is nothing inherently reliable about any scientific claim or theory, since new evidence may be found at any time to overthrow it.  The history of science is littered with examples.   (The second key feature is that anyone may do this contesting; science is not, or rather should  not be, a priesthood.)
One could perhaps defend Atkins’ statement by saying that the abstracted method – first announcement of a claim or hypothesis about the world, then running experiments in the world aiming to falsify the claim, then objective revision or retraction of the claim – may lead to reliable knowledge over the long term.   But, as Paul Feyerabend argued from examination of historical records of scientific disputes, actual living, breathing scientists rarely follow any such method:  they merely use whatever argumentation techniques best suit their material at the time in an attempt to win support for their claims, and they typically maintain their personal support for their own claims despite any contrary evidence.  Given such diversity of actual scientific argumentation practice across disciplines, across time and across issues, I think it only a foolhardy person who would seek to demonstrate that an abstraction from these practices was guaranteed to yield reliable knowledge about the world. (I speak as someone who has tried to do just this, under some severe assumptions as to the types of knowledge and the types of argument used.  See reference below.) 
Second, it is worth noting that Atkins appears to have overlooked other means of achieving knowledge. Pure mathematics, for example, produces new knowledge by means of deductive reasoning, not using anything resembling the scientific method.  Many argue that such knowledge is reliable, since once demonstrated claims cannot be overthrown (at least, not overthrown using the same assumptions and same rules of inference).  Most theories of theoretical physics in the modern era – from Isaac Newton’s theory of gravity, through Einstein’s relativity theories, right up to contemporary string theory and brane theory – are mathematical in kind, developed by mathematicians from their intuitions and using deductive reasoning without recourse to experiment or anything approaching the scientific method. Newton, for instance, famously assumed that the physical laws which governed the motion of planets around the sun also governed the swings of pendulums here on earth, an assumption for which he had not a skerrick of evidence (and nor could he have had), and which is completely counter-intuitive.   We western moderns do not think it counter-intuitive because we have each received a decade of indoctrination at school in the objectively-weird notions of physics since Newton; without talk of mystical (and never fully explained) forces called “gravity” we too would find this assumption obviously without basis. 
In fact, most of our knowledge of physical Nature comes from these mathematical theories, even in Atkins’ own field of chemisty (where the very abstract mathematical theory of groups finds application).  Of course, we aim to test such mathematical theories by means of experiment, but a test is for the purposes of acceptance or rejection of the theory.   Once tested, the knowledge we have of Nature is from the theory, not usually from the test.   Arguably, it is mathematics, not the scientific method, which provides the knowledge we have.  In the case of string and brane theories, for example, no experiments to test these theories have yet been undertaken, and perhaps none could be undertaken even in principle (since the theories concern dimensions of space-time inaccessible to us).   In this case, not only the knowledge but even the acceptance or rejection of the theory, is from the mathematics, and not from something called a scientific method.   (And on what mathematical basis would we accept a mathematical theory of nature?  Perhaps on its elegance or mathematical beauty, or its simplicity, or its profoundness, or its tractability, or its computability.)
It is worth noting here, also, that in many cases, mathematical or computational models in science provide our only means to apprehend the Nature they are intended to model or describe.  We cannot know whether string theory, for example, describes the natural world well or not because we have no other way to apprehend or observe that part of the world it purports to describe.   It it is therefore moot to say that such theories are “reliable” or “effective”, since how could we tell? 
Atkins also ignores knowledge about actions, as distinct from knowledge of facts (know-how, rather than know-what).  For example, our knowledge about technologies and how they work – surely an important part of knowledge about the world – is typically gained not through scientific experiment aiming to test some explicit prior hypothesis, but through building prototypes and exploring the properties of these human artefacts.  This process of creation and exploration is closer to play than to anything a philosopher of science would term the scientific method.   Similarly, an artist’s knowledge of some object (real or imagined) may be gained by drawing or painting it, using drawing as a form of thinking, again an activity very much like play.  To argue that any such knowledge gained by the artist is not knowledge, or perhaps not knowledge about Nature, would be reductionist (and, I think, perverse). 
And Atkins has also ignored, as Norm points out, the insight and knowledge  about the world provided by the humane disciplines – theology, literature, philosophy, etc.   As I have argued before, getting things done in the world requires, inter alia, a knowledge of how people and groups behave and function.   The best source of such knowledge is not science or the scientific method (despite the pretensions of academic social psychology), but literature, TV dramas, and films.
Reference:
P. McBurney and S. Parsons [2001]: Representing epistemic uncertainty by means of dialectical argumentation. Annals of Mathematics and Artificial Intelligence. Special Issue on Representations of Uncertainty. 32 (1-4): 125-169.