Theorems with an extraordinary exception or a small number of sporadic exceptions
The Whitney graph 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: 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.
(A late thanks to Asaf!)
Every automorphism of $S_n$ is inner if $n \neq 6$.
P.S. That $S_6$ has an `essentially' unique outer automorphism is quite a non-obvious fact.
I've posted this answer to other questions, but it's worth repeating:
The topological manifold $\mathbb{R}^n$ has a unique smooth structure up to diffeomorphism as long as $n \neq 4$. However, $\mathbb{R}^4$ admits uncountably many exotic smooth structures.
Edit: Other thoughts:
The h-cobordism theorem holds for $n= 0, 1, 2$ and $n \geq 5$. It is open for $n = 3$ and false (smoothly) for $n=4$.
I've heard that many theorems in number theory only work for field characteristic $\neq 2$ (but my number theory background is sadly lacking).
$(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic iff $n = 1, 2, 4, p^k, 2p^k$, where $p$ is an odd prime, $k \geq 1$. (Granted, there aren't a finite number of exceptions here, but I wanted to mention it anyway.)
Carmichael's Theorem: Every Fibonacci number $F_n$ has a prime factor which does not divide any earlier Fibonacci number, except for $F_1 = F_2 = 1$, $F_6 = 8$ and $F_{12} = 144$.
This is a special case of his more general result: Let $P,Q$ be nonzero integers such that $P^2 > 4Q$, and consider the Lucas sequence $D_1 = 1$; $D_2 = P$; $D_{n+2} = P \cdot D_{n+1} - Q \cdot D_n$. Then all but finitely many $D_n$ have a prime factor which does not divide $D_m$ for any $m < n$; the only possible exceptions are $D_1$, $D_2$, $D_6$ and $D_{12}$.