For a symmetric positive definite Toeplitz matrix it has been established by R. H. Chan and G. Strang [SIAM J. Sci. Statist. Comput. 10, No. 1, 104-119 (1989; Zbl 0666.65030)] that it is possible to apply the conjugate gradient method with circulant preconditioners so that the method converges superlinearly. This is known to depend on whether the eigenvalues of the preconditioned matrix cluster around \(1\). Since analogous results are known for a variety of situations, a natural guess is that a similar result could also be obtained when the matrix is multilevel, which corresponds to Toeplitz matrices associated with Fourier series in several variables. The paper shows that this is never the case, namely that multilevel circulant(-like) preconditioners for multilevel Toeplitz matrices are not superlinear. The results obtained in the paper include as a particular case that there are Hermitian multilevel Toeplitz matrices for which any Hermitian multilevel circulant-like preconditioner is not superlinear.