TITLE: What are some examples of theorem requiring highly subtle hypothesis?
QUESTION [39 upvotes]: I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almost absent from the proof but which are actually hidden behind some really abstract or technical argument. It would be even more interesting if this unnoticed hypothesis was not noted at first but later it had to be added in another paper or publication not because the proof of the theorem were wrong but because the author did not notice that this or that condition was actually playing a role behind the scene and needed to be added. And, finally, an extra point if this hidden hypothesis led to some important development or advance in the area around the theorem in the sense that it opened new questions or new paths of research. This question might be related with this other but notice that it is not the same as I am speaking about subtleties in proof that were not exactly incorrect but incomplete in the sense of not mentioning that some object or result had to be use maybe in a highly tangential way.
In order to put some order in the possible answers and make this post useful for other people I would like you to give references and to at least explain the subtleties that helps the hypothesis to hide at a first sight, expose how they relate with the actual proof or method of proving, and tell the main steps that were made by the community until this hidden condition was found, i.e., you can write in fact a short history about the evolution of our understanding of the subtleties and nuances surrounding the result you want to mention.
A very well known and classic example of this phenomenon is the full theory of classical greek geometry that although correctly developed in the famous work of Euclides was later found to be incompletely axiomatized as there were some axioms that Euclides uses but he did not mention as such mainly because these manipulations are so highly intuitive that were not easy to recognize that they were being used in an argument. Happily, a better understanding of these axioms and their internal respective logical relations through a period of long study and research lasting for millennia led to the realization that these axioms were not explicitly mention but necessary and to the development of new kinds of geometry and different geometrical worlds.
Maybe this one is (because of being the most classic and expanded through so many centuries and pages of research) the most known, important and famous example of the phenomena I am looking for. However, I am also interested in other small and more humble examples of this phenomena appearing and happening in some more recent papers, theorems, lemmas and results in general.
Note: I vote for doing this community wiki as it seems that this is the best way of dealing with this kind of questions.

REPLY [24 votes]: This is not a perfect example because the subtle hypotheses in question were not "unnoticed"; nevertheless I think it fulfills several of your other criteria.  Let us define the "Strong Fubini theorem" to be the following statement:

If $f:\mathbb{R}^2 \to \mathbb{R}$ is nonnegative and the iterated integrals $\iint f\,dx\,dy$ and $\iint f\,dy\,dx$ exist, then they are equal.

The Strong Fubini theorem looks innocent enough, but without any measurability hypotheses, it is independent of ZFC. For example, Sierpinski showed that Strong Fubini is false if the continuum hypothesis holds.
In the other direction, a paper by Joe Shipman investigates a variety of interesting hypotheses that imply Strong Fubini, e.g., RVM ("the continuum is real-valued measurable"), which is equiconsistent with the existence of a measurable cardinal.
Here's another one: Let $\kappa$ denote the minimum cardinality of a nonmeasurable set, and let $\lambda$ denote the cardinality of the smallest union of measure-zero sets which covers $\mathbb{R}$. Then the assertion that $\kappa < \lambda$ implies Strong Fubini.

REPLY [24 votes]: Theorem. Assuming the axiom of choice, the countable union of countable sets is countable.

Proof. Let $\{A_n\mid n\in\Bbb N\}$ be a family of countable sets, and so we can write $A_n$ as $\{a_{n,m}\mid m\in\Bbb N\}$.
Let $A$ be the union, and define $f(a) = 2^n3^m$ such that $n$ is the least such that $a\in A_n$, and $a=a_{n,m}$. Easily, this is an injection so the union is countable.

The trained eye, of course will notice the use of the axiom of choice immediately. We choose an enumeration of each $A_n$. But this is very subtle and usually people will not notice that at first.
And of course, this use of choice is necessary. Indeed, it is consistent that the real numbers are a countable union of countable sets! (Still uncountable, though.)