  We study the Hilbert functions of fat points in P^1 x P^1. If Z is an
arbitrary fat point subscheme of P^1 x P^1, then it can be shown that for every
i and j the values of the Hilbert function H_Z(l,j) and H_Z(i,l) eventually
become constant for l >> 0. We show how to determine these eventual values by
using only the multiplicities of the points, and the relative positions of the
points in P^1 x P^1. This enables us to compute all but a finite number values
of H_Z without using the coordinates of points. We also characterize the ACM
fat points schemes using our description of the eventual behaviour. In fact, in
the case that Z is ACM, then the entire Hilbert function and its minimal free
resolution depend solely on knowing the eventual values of the Hilbert
function.
