  In this work, we introduce a natural notion concerning finite vector spaces.
A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$, which forms a
partial spread, is called almost affinely disjoint if any $(k+1)$-dimensional
subspace containing a subspace from the family non-trivially intersects with
only a few subspaces from the family. The central question discussed in the
paper is the polynomial growth (in $q$) of the maximal cardinality of these
families given the parameters $k$ and $n$. For the cases $k=1$ and $k=2$,
optimal families are constructed. For other settings, we find lower and upper
bounds on the polynomial growth. Additionally, some connections with problems
in coding theory are shown.
