Hang on. Why is this a problem? I don't know if you are referring to a specific case (if so, please elaborate), but why would the following be an impossible scenario:
Organism X lives in Southern Asia.
During a change in climate, conditions favourable to its northward spread occurs, and X happens to spread north until it is at the limits of where it could reproduce. This happens to be Mongolia.
Climate changes again, but more slowly, allowing the Mongolian populations to adapt to local conditions. Eventually, it is better adapted to the Mongolian conditions than to those of Southern Asia.
These conditions happen to be uniform over a large area, spreading from Ussuriland to Portugal, and X gradually comes to spread over this area. Populations in different parts of the range are variously isolated from each other, depending on geographical features and distance. Gradually, these populations adapt to local conditions.
Given sufficient time, the populations of the extreme edges of the range are sufficiently different for us to interpret this (whether morphologically, genetically, or by other methods) as a set of different, but closely related, species. Perhaps one in Southern France is named species X1, the one in Ussuriland and Mongolia X2 and the one in Southern Asia X3.
After having found fossils of all three, as well as other populations, deemed to be intermediate or otherwise related to the group, and analysing them genetically, we feel confident enough in our data to publish them, employing the scenario outlined above.
What part of that is actually impossible? We know that organisms are capable of extending their ranges. We know that organisms, given time, may spread over very large distances, and even, depending on their mode of migration, form disjunct ranges. We know that, again given time, organisms tend to adapt to their local environment (if necessary). We know that is it possible for two allopatric populations to become reproductively isolated from each other -- even for sympatric populations to become effectively reproductively isolated from each other. We know that when this happens, and the isolation is sufficiently complete, adaptations in one part of an organism's range will be independent of adaptations in another part, whether this part is disjunct or not. We know that the local environment of different parts of a given organism's range may be slightly or substantially different.
Where does the impossibility come in? I certainly cannot see it.
A former PhD student here at my department studied this and claimed that he could identify 37 distinct ones (this was about three years ago, I think), although many of them are extremely similar and perhaps better seen as variants of the same concept. Possibly, if these variants are lumped together, your number is more or less accurate.
Of course we do (1), as the concept of "species" is elusive, and we still haven't found a definition that fits all known organisms. Personally, I think de Queiroz is perhaps closest to getting a genuinely universal one but we're not there yet.
As a working biologist, I would rather have different species concepts for different groups of organisms, than trying to apply the same (by necessity arbitrary) standards equally to all groups of organisms. Doing so would create more problems than it solves, and switching between different species concepts is generally not a problem in biology, as long as you are clear about what concept you are applying to your group.
Naturally, even people working on the same group of organisms will have different ideas about where to draw the lines and what criteria to use and so on, but this is very rarely a real problem. I was in precisely one of these discussions the other day, when I proposed to synonymise two species because they are genetically identical but morphologically distinct. A colleague in New Zealand disagreed, and I simply outlined my reasoning for why I thought they should be the same, despite the considerable differences in morphology, and even though we still don't agree, it isn't a problem.
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(1) I am a professional biologist, so I get to use "we" proudly. I might also have no problems being called a "Darwinist", provided the term is adequately explained first. As long as you don't call me an Eichlerist (if he wasn't already dead, I might have wished him to be so that his silliness would stop; but this is a digression).
