If R is a ring of subsets of a set Ω and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ∆ of R is called a Nikodým set for ba (R) if each set {µ α : α ∈ Λ} in ba (R) which is pointwise bounded on ∆ is norm-bounded in ba (R). If the whole ring R is a Nikodým set, R is said to have property (N), which means that R satisfies the Nikodým-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck's property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikodým sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N) ⇔ (wN) holds