\(H\)- and \(Z\)-eigenvalues of an \(n\)-dimensional tensor of order \(m\) were defined by, e.g. L. Qi [J. Symb. Comput. 40, No. 6, 1302–1324 (2005; Zbl 1125.15014)]. In this paper, upper bounds for the spectral radius (i.e., the maximum absolute value of the elements) of each of these spectra are derived for tensors with nonnegative elements. In the case of the \(Z\)-spectrum, the tensor is irreducible and weakly symmetric in the sense of K. C. Chang et al. [Linear Algebra Appl. 438, No. 11, 4166–4182 (2013; Zbl 1305.15027)]. Also \(B\)-eigenvalues defined, e.g. by K.-C. Chang et al. [Commun. Math. Sci. 6, No. 2, 507–520 (2008; Zbl 1147.15006); errata ibid. 10, No. 3, 1025 (2012)] are considered (i.e., the analog of generalized eigenvalues for a couple of matrices generalized to a couple \((A,B)\) of tensors) and also here a bound for the corresponding spectral radius is obtained.