Let G be one of the groups SLn(ℂ), Sp2n(ℂ), SOm(ℂ), Om(ℂ), or G2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙN is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.