It is well known that for a number of structures, their existence is equivalent to the existence of a projective plane for a given order. Some of them depend on more than one parameters, which means that there can be several ways of measuring "how close" one can approach a (conjecturally non existing) projective plane for a given order $n$ with more than one prime divisor.
To illustrate this, take e.g. the biggest number $a(n)$ of mutually orthogonal latin squares (MOLS) of order $n$. For prime order, we have trivially $a(n)=n-1$, and moreover non isomorphic sets of $n-1$ MOLS correspond to non isomorphic projective planes of order $n$. On the other hand, it is well known that for $n=6$, there are not even two MOLS, i.e. $a(6)=1$, further that $2\le a(10)\le 6$. This survey is not very recent but gives a lot of lower bounds. The well-known fact that $a(12)\ge5$ means e.g. that for $n=12$, one can come "much closer" (in a certain sense) to a projective plane than for $n=6$.
A different way of measuring is invoked in this recent thread.
Which other structures allow to "measure" how close one can get?
(I would have thought that such a collection already exists in MO, but I couldn't find any.)