Question: The [Whitney graph isomorphism theorem](https://en.wikipedia.org/wiki/Line_graph#Whitney_isomorphism_theorem) gives an example of an extraordinary exception: a very general statement holds except for **one** very specific case.

Another example is the [classification theorem for finite simple groups](http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Statement_of_the_classification_theorem): a very general statement holds except for **very few (26)** sporadic cases. 

I am looking for more of **this kind** of theorems-with-not-so-many-sporadic-exceptions 

> 
> (***added:***) where the exceptions don't come in a row and/or
>  in the beginning - but are scattered truly [sporadically](https://en.wiktionary.org/wiki/sporadically).
> 
> 
> 

(A late thanks to Asaf!)

Answer: 
There are several theorems in topology and geometry that hold for manifolds of all dimensions save 3 and sometimes 4. For example, of Euclidean space except $\mathbb{R}^3$ and sometimes $\mathbb{R}^4$. There are also some conjectures proved for $\mathbb{R}^n$, n ≠ 3 (resp {3,4}), but which are open questions for n = 3 ({3,4}). Wikipedia describes it thus:

> 
> [T]he cases $N = 3$ or $4$ have the richest and most difficult geometry and topology. There are, for example, geometric statements whose truth or falsity is known for all N except one or both of 3 and 4. N = 3 was the last case of the Poincaré conjecture to be proved.
> 
> 
> 

Using the [regular polytopes](http://en.wikipedia.org/wiki/List_of_regular_polytopes) as an example:

* There are exactly 3 regular (convex) polytopes, the n-simplexes, -cubes, and -orthoplexes which exist in all dimensions, *except* 3 and 4, which have several additional Platonic solids and regular polychora without higher analogues (ignoring the trivial cases of dimensions 1 and 2).
* There are no regular non-convex polytopes *except* in dimensions 2, 3, and 4.

Further,

* [Moise's theorem](https://en.wikipedia.org/wiki/Moise%27s_theorem) holds only in 3 dimensions.
* There is exactly 1 differentiable structure on $\mathbb{R}^n$, up to diffeomorphism, [*except* for n = 4](https://en.wikipedia.org/wiki/Exotic_R4), which has infinitely many. See ["Bizarre Fact 48"](http://www1.maths.leeds.ac.uk/~speight/teaching/diffgeom/dgnotes.pdf).

In a sense, 3- and 4-dimensional spaces are privileged. This has implications in the philosophy of physics: why did space ([apparently](https://en.wikipedia.org/wiki/Extra_dimensions)) have [3 spatial dimensions rather than 2 or 527](https://en.wikipedia.org/w/index.php?title=Spacetime&oldid=508323437#Privileged_character_of_3.2B1_spacetime)?

See also [3-manifold](https://en.wikipedia.org/wiki/3-manifold), [special phenomena of 4-manifolds](https://en.wikipedia.org/w/index.php?title=4-manifold&oldid=496192106#Special_phenomena_in_4-dimensions), [low-dimensional topology](https://en.wikipedia.org/wiki/Low-dimensional_topology).
The number of partitions of $n\geq 0$ into parts congruent to $\pm 1,
\pm 4, \pm 5, \pm 6, \pm 7, \pm 9, \pm 11, \pm 13, \pm 16, \pm 21, \pm
23, \pm 28$ (mod 66) is equal to the number of partitions of $n\geq 0$
into parts congruent to $\pm 1, \pm 4, \pm 5, \pm 6, \pm 7, \pm 9, \pm
11, \pm 14, \pm 16, \pm 17, \pm 27, \pm 29$ (mod 66) except for
$n=13$. See [here](https://qseries.org/fgarvan/papers/shifted.pdf) and [here](https://arxiv.org/pdf/math/0605317.pdf) for this and
many similar results.