The combination of searches performed by the CDF and D0 collaborations at the Fermilab Tevatron Collider for neutral Higgs bosons produced in association with b quarks is reported. The data, corresponding to 2.6 fb-1 of integrated luminosity at CDF and 5.2 fb-1 at D0, have been collected in final states containing three or more b jets. Upper limits are set on the cross section multiplied by the branching ratio varying between 44 pb and 0.7 pb in the Higgs boson mass range 90 to 300 GeV, assuming production of a narrow scalar boson. Significant enhancements to the production of Higgs bosons can be found in theories beyond the standard model, for example in supersymmetry. The results are interpreted as upper limits in the parameter space of the minimal supersymmetric standard model in a benchmark scenario favoring this decay mode.