In the definition of a Kan fibration (on nlab), i.e. for a map $\pi:Y\to X$ of simplicial sets the inclusion of any horn into $Y$ always lifts to an inclusion of the filled in horn if that filled in horn includes into $X$, it seems hard to imagine, at least geometrically, a counterexample. Is there some intuitive, visual way to see under what conditions this might not happen? Also, if my definition is not correct, please edit!
Thanks!
(I'm sure by now you know of many examples, but in case someone else wanted an answer to this question...)
For one, any non-Kan complex has the property that $K \rightarrow \ast$ is not a Kan fibration. For example, any horn $\Lambda^n_i$ fails to be a Kan complex (for obvious reasons), and the nerve of any category that's not a groupoid will fail to be a Kan complex (this includes, as a special case, $\Delta^n = N([n])$).
Non-point examples include: Most functors $\mathcal{C} \rightarrow \mathcal{D}$ between categories do not induce Kan fibrations on nerves $N\mathcal{C} \rightarrow N\mathcal{D}$ (this happens if and only if you have a category both fibered and cofibered in groupoids!) [Also, to your point about different model structures, these are examples of "inner fibrations" that are not Kan fibrations... if you have a category cofibered in groupoids, you get "left fibrations" that are not Kan fibrations, etc.)
Basically most maps arising in higher category theory are not Kan fibrations... This is usually a special thing and when it happens you're happy.