Supersymmetry has been often invoqued as the new physics that might reconcile
the experimental muon magnetic anomaly, a_mu, with the theoretical prediction
(basing the computation of the hadronic contribution on e^+ e^- data). However,
in the context of the CMSSM, the required supersymmetric contributions (which
grow with decreasing supersymmetric masses) are in potential tension with a
possibly large Higgs mass (which requires large stop masses). In the limit of
very large m_h supersymmetry gets decoupled, and the CMSSM must show the same
discrepancy as the SM with a_mu . But it is much less clear for which size of
m_h does the tension start to be unbearable. In this paper, we quantify this
tension with the help of Bayesian techniques. We find that for m_h > 125 GeV
the maximum level of discrepancy given current data (~ 3.3 sigma) is already
achieved. Requiring less than 3 sigma discrepancy, implies m_h < 120 GeV. For a
larger Higgs mass we should give up either the CMSSM model or the computation
of a_mu based on e^+ e^-; or accept living with such inconsistency
