We study the Hilbert polynomials of finitely generated graded algebrasR, with generators not all of degree one (i.e. non-standard). Given an expressionP(R,t)=a(t)/(1-tl)nfor the Poincare series ofRas a rational function, we study for 0 ≤i≤lthe graded subspaces ⊗kRkl+i(which we denoteR[l;i]) ofR, in particular their Poincaré series and Hilbert functions. We prove, for example, that ifRis Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, ifRis also a domain then these Hilbert polynomials have the same leading coefficient.