  In the Minimal Supersymmetric Extension of the Standard Model (MSSM) the
higgsino mass parameter $\mu$ appears both in the masses of the Higgs bosons
and in the neutralino mass matrix. Electroweak finetuning therefore prefers
small values of $|\mu|$. On the other hand, bino--like neutralinos make a good
dark matter candidate. We show that current direct search limits then impose a
strong lower bound on $|\mu|$, in particular for $\mu > 0$ or if the masses of
the heavy Higgs bosons of the MSSM are near their current limit from LHC
searches. There is therefore some tension between finetuning and neutralino
dark matter in the MSSM. We also provide simple analytical expressions which in
most cases closely reproduce the numerical results.
