We extend the class of measures defined on orthomodular lattices or posets, orthoalgebras, and on effect algebras for which the Nikodỳm boundedness theorem holds. We present variants for finitely additive measures and completely additive measures. As a corollary, we obtain Nikodỳm's boundedness theorem for measures defined on the system $L(H)$ of all closed subspaces of a real or complex Hilbert space $H$. This result will be proved without using the Bair category theorem.