  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.
