Let $\Sigma$ be a ring of sets, $X$ a normed space, $\mu_\alpha:\Sigma\to X$ ($\alpha\in\Lambda$) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family $\{\mu_\alpha\}$$\{\mu_\alpha\}$ is uniformly bounded on $\Sigma$ if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a~part of some set in~$\Sigma$. An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a~family of triangular functions is preserved in passing from a~ring to the $\sigma$-ring it generates.