We study the Hilbert functions of fat points in ℙ1 × ℙ1. If
Z⊆ ℙ1 × ℙ1 is an arbitrary fat point scheme, then it can be shown that for every iand jthe values of the Hilbert function HZ( l, j) and HZ( 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 ℙ1 × ℙ1. This enables us to compute all but a finite number values of HZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case that Z⊆ ℙ1 × ℙ1 is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.