Parameterized complexity from P to NP-hard and back again
One field with lots of non-monotonicity of problem complexity is property testing. Let $\mathcal{G}_n$ be the set of all $n$-vertex graphs, and call $P \subseteq \mathcal{G}_n$ a graph property. A generic problem is to determine whether a graph $G$ has property $P$ (i.e. $G \in P$) or is `far' from having property $P$ in some sense. Depending on what $P$ is, and what kind of query access you have to the graph, the problem can be quite difficult.

But it is easy to see that the problem is non-monotone, in that if we have $S \subset P \subset T$, the fact that $P$ is easily testable does not imply either that $S$ is easily testable or that $T$ is.  

To see this, it is enough to observe that $P = \mathcal{G}_n$ and $P = \emptyset$ are both trivially testable, but that for some properties, there exist strong lower bounds.