This paper is devoted to the following problem: given two commuting bounded Boolean algebras \({\mathcal M}\) and \({\mathcal N}\) of commuting projections of a Banach space \(X\), when is the Boolean algebra of projections generated by \({\mathcal M}\) and \({\mathcal N}\) bounded? For simplicity, we shall call this property \(B({\mathcal M}, {\mathcal N})\). This problem goes back to the fifties (1954) when J. Wermer showed that \(B({\mathcal M}, {\mathcal N})\) always happens when \(X\) is a Hilbert space. Subsequent works focussed on finding classes of Banach spaces in which \(B({\mathcal M}, {\mathcal N})\) happens for every commuting pair \(({\mathcal M}, {\mathcal N})\): \(L^p\)-spaces for \(1 \leq p < \infty\) (C.~A.\ McCarthy), Grothendieck spaces with the Dunford--Pettis property [\textit{W.~Ricker}, Bull.\ Lond.\ Math.\ Soc.\ 17, 268--270 (1985; Zbl 0584.47033)], hereditarily indecomposable Banach spaces [\textit{W.~Ricker}, Acta Sci.\ Math.\ 59, No. 3--4, 475--488 (1994; Zbl 0836.47023)], Banach lattices, closed subspaces of \(p\)-concave Banach lattices (\(p\) finite), complemented subspaces of \({\mathcal L}^\infty\) spaces, Banach spaces with local unconditional structure (l.u.st.) [\textit{T. A. Gillespie}, J.\ Funct.\ Anal.\ 148, No.~1, 70--85 (1997; Zbl 0909.46017)]. In this paper, the authors point out that a possible property of the Boolean algebras themselves plays an important role: the notion of \(R\)-boundedness. A non-empty subset \({\mathcal T}\) of operators on a Banach space \(X\) is said to be \(R\)-bounded if there exists a constant \(M\geq 0\) such that \[ \bigg(\int_0^1 \Big\| \sum_{j=1}^n r_j(t) T_j x_j \Big\| ^2\,dt\bigg)^{1/2} \leq M\,\bigg(\int_0^1 \Big\| \sum_{j=1}^n r_j(t) x_j\Big\| ^2\,dt\bigg)^{1/2} \] for all \(T_1,\ldots, T_n\in {\mathcal T}\), \(x_1,\ldots, x_n \in X\), \(n\geq 1\). The authors show that when at least one of the Boolean algebras \({\mathcal M}\) or \({\mathcal N}\) is \(R\)-bounded, then \(B({\mathcal M}, {\mathcal N})\) is satisfied, and that when \(X\) has property \((\alpha)\), every bounded Boolean algebra of projections is \(R\)-bounded. A Banach space \(X\) has property \((\alpha)\) if there exists a constant \(\alpha \geq 0\) such that \[ \int_0^1\int_0^1 \Biggl\| \sum_{j=1}^m \sum_{k=1}^n \varepsilon_{jk} r_j(s) r_k(t) x_{jk}\Biggr\| ^2 \,ds\,dt \leq \alpha^2 \int_0^1\int_0^1 \Biggl\| \sum_{j=1}^m \sum_{k=1}^n r_j(s) r_k(t) x_{jk}\Biggr\| ^2\,ds\,dt \] for every \(x_{jk}\in X\) and every choice of signs \(\varepsilon_{jk}=\pm 1\), \(m,n\geq 1\). Property \((\alpha)\) was introduced by \textit{G.~Pisier} [Compos.\ Math.\ 37, 3--19 (1978; Zbl 0381.46010)] and is satisfied by all Banach spaces with l.u.st.\ and finite cotype; for Banach lattices, property \((\alpha)\) is equivalent to have a finite cotype or to be \(p\)-concave for some finite \(p\). More generally, GL-spaces (i.e., spaces with the Gordon--Lewis property) also have property \((\alpha)\) if and only if they have finite cotype. The authors show, however, that in any GL-space, without any assumption on the cotype, the property \(B({\mathcal M}, {\mathcal N})\) happens for every pair \(({\mathcal M}, {\mathcal N})\). All the above results are given with explicit bounds on the norms of the Boolean algebras. The authors also prove that a Banach lattice \(E\) has finite cotype (or, equivalently, property \((\alpha)\)) if and only if \(E\) is Dedekind \(\sigma\)-complete and the particular Boolean algebra \({\mathcal B}(E)\) of all band projections is \(R\)-bounded. Finally, they show that in any Banach space the strong operator closure of any \(R\)-bounded Boolean algebra of projections is always Bade-complete.