Consider the partial \(n\times n\) matrix \(F\) with bounded Hilbert space operator entries, where the lower triangular entries are specified and the strictly upper triangular entries are to be determined. Any choice of the strictly upper triangular entries provides a completion or extension of \(F\). The approach of this paper is to reduce the analysis of \(n\times n\) case to iterations of \(2\times 2\) completions. Let \(0 \min\{ \nu,\mu \}\). The operator matrix \({{b\;x} \choose {a\;c}}\) is called a \((\nu, \mu, \delta)\)-central completion if \(|{b\choose a}|\leq\nu\), \(|(a, c)|\leq \mu\) and \(x=- b(\delta^2 I-a^* a)^{-1} a^* c\). The authors characterize such central completions and analyze the upper bound for such central completions. If \(F\) is an \(n\times n\) operator matrix, then its central completion is obtained by determining as above the central completion of the first two columns of \(F\), then of the first three columns, and so on. They prove that the upper bounds are attained for \(2\times 2\) case and cannot be attained for \(n\times n\) cases with \(n>2\).