Purely because I didn't see any combinatorial example:

The Blow-up Lemma says that if you have a regular partition of a graph $G$, and a bounded-degree subgraph $H$ on the same number of vertices, then you can embed $H$ in $G$ if you 'should be able to', that is you can find a homomorphism from $H$ to the reduced graph of $G$ (put edges between parts of the partition corresponding to dense regular pairs) which maps the right number of vertices of $H$ to each part of $G$. Provided that you have 'super-regularity', which more or less means no cheating by having isolated vertices. If you unpack the vagueness here you get a reasonable list of conditions.

If you want to go further, try repeating this in sparse graphs. Now $G$ will have to be a subgraph of some random or pseudorandom graph, otherwise (as the OP knows well) the whole theory falls apart, but in addition there are two other ways to 'cheat' (two quite distinct ways of interfering with vertex neighbourhoods) and you have to exclude these to have a Blow-up Lemma.

I think this kind of thing is pretty ubiquitous in mathematics, actually: you have a general idea that in all nice situations X will be true. But when you try to make it precise, you discover a bunch of distinct nasty situations in which X fails, so you exclude them and then you have a theorem with many hypotheses.