It doesn't help that our Linnaean system was established by, well, Linnaeus. Brilliant though he was, he lived and produced his great work before Darwin developed the theory of evolution. I think I recall he was a special creationist - at that time creation was as good an explanation as any.
We might infer then that he believed species were essentially immutable. This makes for a disconnect between the discrete Linnaean taxonomic system and the continuous nature of species being categorized.
Consider the following:
We start with species 'A' someplace way back in time. Through time, the descendents of this species branch all over the place - into a bush, not a tree. Let's follow one of the myriad paths from this relative 'root' to the currently existing leaf species. Say 'Z' is a modern descendent.
So we have A begat B begat C begat .... Z
Now, if the delta T (the elapsed time) has been a long time (on the order of 100s of thousands or millions of years), then A and Z are probably not the same species - they might not even be the same genus. (Note that A and Z *might* be the same species, because there's nothing I'm aware of in evolutionary theory that says that species have to die out.)
However, in each case, A is the same species as B is the same species as C is the same species as D, and so on. Each successor may or may not be assimilating some slight mutation into the gene pool. Eventually there comes a point where, say, X is no longer of the same species as A, even though A is the same species as W and W is the same species as X.
Let the string '<=>' mean 'is the same species as' and let the string '</=/>' mean 'is not the same species as.'
The upshot of my point is that you can have a case where A <=> B and B <=> C, but A </=/> C. In fact, there must have been many such cases. Mathematically we would say that the relationship 'is the same species as' is not transitive.
(It's important to emphasize that this is a thought experiment. I don't think it's possible to look at a fossil, for example, and say with certainty that it was a direct ancestor species of some existing species. The best we might say is that it appears to be at least a cousin, and 'might' be a direct ancestor. This is part of the reason why terms like "missing link" are nonsensical.)
I'm not sure whether a biologist would buy into any of this, btw. But it seems intuitively obvious.
To summarize:
Cause of Problem:
The objects (species) being categorized constitute a continuum, while the taxonomic system assumes discrete units.
Result:
There will be items among the continuum (species) that are putatively difficult to classify.
k
