  We establish that matroids characterized by the Tutte polynomial
$\sum_{i,j\ge 0}t_{i,j}x^iy^j$ with coefficients $t_{i,j}$ vanishing for
$(i,j)\ge (k,l)$ precisely coincide with $(k,l)$-uniform matroids. This
characterization implies that almost $(k,l)$-uniform matroids are exactly
matroids with $t_{k,l}\le 1$ and $t_{i,j}=0$ if $(i,j)>(k,l)$. We also
characterize excluded minors of almost $(k,l)$-uniform matroids in terms of
Tutte polynomial coefficients. Finally, we construct an infinite family of
excluded minors of almost $(k,l)$-uniform matroids which extend previously
known cases of almost uniform and almost paving matroids.
