Eliminating hypotheses is, in general, an inefficient form of inquiry. It only becomes valuable under certain conditions, namely those that have already yielded a reasonable basis for understanding how different hypotheses are relvant and capable of being eliminated in a reasonably clean way.
Arius asked if I could expand on the point, and I'm finally at the point where I have a bit of time to do so. My thoughts on the subject are partially inchoate, but I think we can still go some ways to laying out the conditions under which elimination of hypotheses is an effective form of inquiry.
It's useful, I think, to begin with the concept of a phenomenon. I use the term in Ian Hacking's sense; he didn't invent that sense, but has done the most in recent times to emphasize its importance. We learn about the world from experience. That's almost a truism. But it's important to understand the sense in which it is true. It's not from individual perceptions so much as built-up banks of perceptions that we learn about the world; we need not just a datum but a store of data. Out of this collected experience we begin to recognize intelligible patterns and structures, and this is how we understand the world. But one of the key points here, which is easy to overlook if you don't think it through, is that not just any course of experience will do -- to build the bank of experience we need stability, repeatability, definite pattern. We need phenomena: something that recognizably comes again and again.
The tricky thing is that, if we want to go beyond what we need to know about the world for purely practical purposes, the world is not quite forthcoming. Happenings are common enough, but phenomena are rare; most recurrences in nature are approximate. The theme we need to pin down is there, but it undergoes endless variations, so many that it's difficult to get precise enough about the theme that we actually can say anything about underlying causes. Scientific inquiry consists chiefly in the making and studying of phenomena, and the history of science is heavily constrained by the availability of phenomena to study. There is only one science that has relatively clear cases of phenomena open to everyone's view, and that is astronomy. The lights in the sky are much the same night after night, and of those that are not -- the wandering stars, the planets -- some are sufficiently regular that if you observe them long enough you can see clear patterns. Thus it is astronomy that develops first among the sciences. It is not easy; the basic phenomena are available to be observed but the observing of them so that they are usable as phenomena is often a long, difficult, time-consuming process. The orbits of the planets, despite their variations, are far more regular than most of what we find around us, but it took so many centuries to find ways to measure them, track them, record them with the accuracy needed to draw out the intelligible structure of these phenomena! And those are the phenomena that are largely open and accessible, just waiting to be measured; most phenomena have to be created, isolated out from the flux, educed by carefully engineered situations. The order in which sciences appear is largely influenced by the ease with which phenomena are created, so that, for instance, large families of basic mechanical phenomena are identified fairly early, while sciences like chemistry, biology, psychology, have to find ways to peel away much more complicated forms of variability. I say that the history is 'largely' influenced by this because, of course, a field of scientific inquiry never deals with only one kind of phenomena; with proper instruments you can isolate astronomical phenomena that are not nearly as obvious as the basic phenomena and thus show up much later in the history of science. Much of science deals with the creation of new phenomena: an experiment is nothing other than an attempt, through a contrived situation, to make a real phenomenon that can be directly or indirectly experienced.
I suggest that the existence of accessible phenomena is the primary condition for effective inquiry by elimination of hypotheses. In everyday practical life it is rarely useful to try to figure things out by elimination of hypotheses. We do use eliminative arguments, but they usually require no hypothesizing because they are based on things that are old hat -- we already know the routes and we're just deciding which one was taken. This is different from an eliminative hypothetical inference because such an inference is an attempt to discover a route in the first place. We do use educated guesses, but they differ from hypotheses in that they just need to be good enough; we do try to find confirmations for our guesses, but we usually are fine with just one or two. And it's easy to see why this is the most reasonable way of proceeding in everyday practical life -- we don't have the time, or the resources, or the interest, in creating phenomena for everything.
It's also the case that even in purely theoretical realms there's no use hypothesizing if your hypotheses are just guesses. From expositions of scientific method that one reads about in elementary and high school textbooks people often come away with the idea that hypothesizing is just the scientific word for guessing. But even to reach the point of having a recognizable hypothesis, we need to have a fairly good sense of what we're dealing with. There's no use trying to make a hypothesis if there is an infinite field of possible answers. What differentiates hypotheses from guesses, even educated ones, is that good hypotheses are things that are actually possible solutions, not mere approximations. In effective inquiry, formulation of a hypothesis is the attempt to identify a really possible alternative in the field of possibilities based on our acquaintance with the phenomena. The farther we get away from phenomena, the more our attempts to hypothesize begin to look like guesses in the dark. Guesses can be entirely reasonable and rational -- the notion that reasonable people never guess is thoroughly absurd. But when you guess and it turns out to be good enough, you haven't tested a hypothesis; you made a reasonable but still partly arbitrary decision and turned out to be lucky. A thriving intellectual life will take advantage of such lucky guesses whenever it can; but it's not hypothesizing in any proper sense of the term.
We also need phenomena for the elimination of hypotheses. Eliminating hypotheses is difficult; crucial experiments only arise under extraordinary circumstances, and even then you want to be able to do them more than once just to be sure. The more variable our experience, the less sense it makes to try to do any elimination; we could be at the eliminative process for centuries or millenia, for all we know. And, indeed, the history of science shows that that is sometimes how long it takes if you don't have phenomena in hand. You need something that can help you say, with reasonable certainty, that a given hypothesis is wrong; this requires phenomena.
So phenomena are a necessary condition: if you don't have them, elimination of hypothesis, which is time- and resource-intensive, is not an efficient way to think about the world. The fact that you have phenomena, however, does not guarantee that you can effectively eliminate hypotheses. You need not just the phenomena but some notion of how they are already structured in order to have an idea both what kinds of hypotheses might be relevant and how they might be eliminated using the phenomena. This means that by the time we've reached the stage where it makes sense to formulate hypotheses in the first place our inquiry has already discovered something about the world -- something basic, no doubt, but something real. We've created or isolated out a phenomenon and recognized it for what it is -- this requires that we be able to recognize its features, see something of its patterns, and see that they are regular, stable, repeatable. This kind of knowledge admits of degrees. And reaching a certain degree is needed before it makes much sense to try to hypothesize and eliminate hypotheses.
This, incidentally, is the grain of half-truth in the Popperian thesis that hypotheses must be falsifiable. What is actually true is that in order to eliminate hypotheses effectively we must be able to identify phenomena with sufficient precision that we can correct for errors in our immediate conclusions about them; without such correction we can never be sure that we've actually eliminated anything rather than been fooled by some contamination or bias in the experiment. To the extent that science involves the elimination of hypotheses it needs the hypotheses to be genuinely eliminable. But this is, in fact, nothing to do with the hypotheses themselves; it has to do with the phenomena we have on hand.
Even when we are at the stage of hypothesizing, of course, we may not be in a position to do any elimination; it may be more reasonable for that stage of inquiry to handle our hypotheses less rigorously by focusing comparing them to analogous cases or trying to find the simplest hypothesis that saves the phenomena that we are currently able to recognize. It takes a lot to get things to the point where elimination of hypotheses is a progressive form of inquiry rather than a sort of random search in field of fog.