To a sequence \(a=\{a_J\}\) indexed by dyadic intervals \(J\) one can associate the square function \(S_\alpha(a)(x)=(\sum_{x\in \alpha J} | a_J| ^2/| J|)^{1/2}\). Here \(\alpha J\) is the interval with the same center and \(\alpha\) times the sidelength of \(J\). When \(\alpha=2\) one simply writes \(S(a)(x)\). A second maximal function can be defined in terms of the wavelet projection \(\Lambda_n(a)(x) =\sum_{| J| >2^{-n}} a_J \psi_J(x)\). When the wavelets \(\psi_J\) are generated by a multiresolution analysis, this is just the projection of \(\sum a_J\psi_J\) onto the \(n\)th multiresolution space. Then one can define \(N\Lambda(a)(x)=\sup_n \sup_{y\in J_n(x)}| \Lambda_n(y)| \). Here \(J_n(x)\) is the unique dyadic interval of length \(2^{-n}\) containing \(x\). Then \(N\Lambda\) is essentially the same as a nontangential maximal function and one expects the same sort of good-\(\lambda\) inequalities to govern \(S_\alpha\) and \(N\Lambda\) as those corresponding to the classical Littlewood-Paley square functions and standard nontangential maximal functions. Indeed, the main results (Theorems 3 and 4) are precisely such analogues. Theorem 3 says that \(S(a)\) cannot be small, too often, when \(N\Lambda(a)\) is large, that is, \[ | \{x\in\mathbb{R}:N\Lambda(a)(x)>k\lambda,\;S(a)(x)\lambda\}| \] for some fixed \(k>1\), provided \(\varepsilon>0\) is sufficiently small. Theorem 4 similarly says that \(N\Lambda(a)\) cannot be too small, too often, when \(S_\alpha(a)\) is too large. Here \(\alpha\) depends on the support of the wavelet.