Given $ax + b = 0$ with $a \neq 0$, I can find a formula for $x$. In particular, we have $x = -b/a$.
Similarly, for $ax^2 + bx + c = 0$ with $a \neq 0$, I can also find a formula for $x$. In particular, the quadratic formula gives all possible values for $x$ (there can be up to $2$).
For the cubic case, one can again come up with an analogous formula, which gives all possible values for $x$ (there can be up to $3$). Probably you see where this is going...
For the quartic case, we have a formula (i.e., using the four operations and radicals) to identify the different possible values of $x$ (here, indeed, there can be up to $4$).
By analogy, one would really expect there to be a "quintic formula." But there's not (in a strong sense, i.e., not just because neither you nor I can think of one), and the reasoning by analogy goes wrong.
(Meanwhile, the idea of "possible values" needs to be made precise, since you might end up allowing for complex values of $x$; this notion does generalize nicely, i.e., a polynomial with degree $n$ can have up to $n$ distinct complex roots.)
Of course, the "reasoning by analogy gone wrong" described above is frequently mentioned early on in a class on, say, Galois Theory to motivate its study.
(And speaking of classical problems: Similar comments could be made about trisecting an angle...)
Below is an example that is a bit simpler to explain in full.
Assume you do not know if a two digit number is prime.
Here is a primality test:
First, memorize the twelve by twelve times table.
Second, check if the given number is even (look at units digit), a multiple of three (check if the digital sum is a multiple of three), or a multiple of five (look at units digit).
Third, if the number does not appear in the times table and has no obvious factors of 2, 3, or 5, then it must be prime.
Now, this approach to finding primes does not work so well for larger numbers.
Moreover, it misses a two digit number: 91, the product of 7 and 13.
(Quite a number of students will misidentify 91 as prime; try presenting it in a list of numbers to a student, and ask him or her to circle all the primes given some reasonable time constraint.)
Of course, for numbers through 100 one needs to check prime factors up to its square root, i.e., 2, 3, 5, and 7. But the tricks for 2, 3, and 5 are very simple, and those of us who memorized twelve by twelve tables get the multiples of 7 up through 84 (hence the missed 91, whereas 98 is even).
Still, I think there is a nice example in here somewhere, and I think it is related to the sense that multiplication tables contain "all the products" in a stronger sense than they really do. For example, the ten by ten table is missing all primes from 11 on. As for composites, it is missing plenty, starting at 22; it just so happens that the primality test described above is effective for all two-digit numbers besides 91.