We study how large the rate of the lepton-flavor violating Higgs decay h->tau mu can be in the (R-parity conserving) MSSM. We make no assumptions, such as universality or alignment, about the flavor structure of the MSSM. We only assume that all couplings and, in particular, the trilinear scalar ones, are perturbative. We take into account lower bounds on the bino and slepton masses from tau->mu gamma and h->gamma gamma. We find that the present bound on BR(h->tau mu) can be saturated only in a highly fine-tuned region of parameter space with a very specific pattern of slepton-mass hierarchy and mixing, heavy higgsinos and bino, and a second Higgs doublet that is lighter than the rest of the supersymmetric spectrum.