  A classic theorem in combinatorial design theory is Fisher's inequality,
which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise
intersections of size $\lambda$ can have at most $n$ non-empty sets. One may
weaken the condition by requiring that for every set in $\mathcal F$, all but
at most $k$ of its pairwise intersections have size $\lambda$. We call such
families $k$-almost $\lambda$-Fisher. Vu was the first to study the maximum
size of such families, proving that for $k=1$ the largest family has $2n-2$
sets, and characterising when equality is attained. We substantially refine his
result, showing how the size of the maximum family depends on $\lambda$. In
particular we prove that for small $\lambda$ one essentially recovers Fisher's
bound. We also solve the next open case of $k=2$ and obtain the first
non-trivial upper bound for general $k$.
