This paper is motivated by a theorem of \textit{H. Donnelly} and \textit{C. Fefferman} [Ann. Math., II. Ser. 118, 593-618 (1983; Zbl 0532.58027)] stating (in part) that if \(\Omega\) is a smooth, bounded, strictly pseudoconvex domain in \(\mathbb{C}^n\), and if \(p+q\neq n\), then there are no non-trivial \((p,q)\) forms on \(\Omega\) that are square-integrable and harmonic with respect to the Bergman metric. An analogous result is known for bounded symmetric domains. The author considers the case of a bounded domain \(\Omega\) in \(\mathbb{C}^n\) that admits a smooth compact quotient by a discrete torsion-free subgroup of the group of holomorphic automorphisms of~\(\Omega\). Also the author restricts attention to forms that are square-integrable with respect to the weight function \(1/d(z)^s\), where \(d(z)\) denotes the Euclidean distance from \(z\) to the boundary of \(\Omega\). The main result is that when \(p+q\neq n\), there are no non-trivial weighted square-integrable harmonic forms when \(s\)~is sufficiently large: namely \(s>n\); or more generally \(s>r_2/r_1\), where the positive real numbers \(r_1\) and~\(r_2\) are defined in terms of the Bergman kernel function \(K(z,z)\) on the diagonal by \(r_1=\sup \{r:\) there exists a positive constant \(C_1\) such that \(K(z,z)\geq C_1/d(z)^r\) for all \(z\) in \(\Omega\}\) and \(r_2=\inf \{r:\) there exists a positive constant \(C_2\) such that \(K(z,z)\leq C_2/d(z)^r\) for all \(z\) in \(\Omega\}\).