Edit:
Since I had some trouble making my previous question precise without diving into details about the origin of the homological objects I'm interested in, let me ask a more open-ended question:
Are there any known examples of the following setting:
A subclass $\mathfrak N$ of an Abelian category $\mathfrak C$ such that $\mathrm{Ext}^2(A,B)=0$ for all pairs of objects $A$, $B$ in $\mathfrak N$, but with infinitely many objects of $\mathfrak N$ having projective dimension greater than $1$?
Original question:
Given an Abelian category $\mathfrak C$ and a subclass $\mathfrak N\subset\mathfrak C$ of "nice" objects, I would like to prove $\mathrm{Ext}^2(A,B)=0$ for all $A,B\in\mathfrak N$.
This would be easy if I knew that all objects in $\mathfrak N$ had projective or injective dimension at most $1$. In this case, I would use that one of the variables is very nice (and also get a stronger result). Let's assume that objects in $\mathfrak N$ are, in general, not nice enough to ensure this.
Question: Are there any conditions for $\mathrm{Ext}^2(A,B)=0$ using that $A$ and $B$ are rather nice (without $A$ or $B$ being very nice)?
Edit: A more concrete description of the objects in the situation I am motivated by: Objects in $\mathfrak C$ are modules over a certain finite category, that is, a bunch of Abelian groups with a bunch of homomorphisms between them, which fulfill certain relations. These modules will often have infinite projective dimension. Objects in $\mathfrak N$ have several special properties: various sequences are exact, certain maps vanish, certain groups are free etc. Objects in $\mathfrak N$ have finite projective dimension.