  We study the Hilbert function and the graded Betti numbers of almost complete
intersection artinian algebras. We show that that every Hilbert function of a
complete intersection artinian algebra is the Hilbert function of an almost
complete intersection algebra. In codimension $3$ we focus on almost complete
intersection artinian algebras whose Hilbert function coincides with that of a
complete intersection defined by $3$ forms of the same degree. We classify all
the possible graded Betti numbers of such algebras and we specify what
cancellations are allowed in a minimal graded free resolution.
