In this work, we present interesting case studies that lead to new and deeper results on fast convergence of reduced-rank conjugate gradient (RRCG) Wiener filters (WF), for applications in communications. and sensor array signal processing. We discover that for signal modes with a specially structured Gram matrix, which induces L groups of distinct eigenvalues in the data covariance matrix, a fast and predictable convergence, in at most L steps, can be achieved when the RRCG WF is used to detect, and/or to focus on, the desired signal mode. For such applications, given knowledge of the repeated eigenstructure of the Gram matrix of signal modes or of the measurement covariance matrix, a RRCG Wiener filter, of at most rank L, delivers the same performance as the full-rank Wiener filter. Typically L is much less than the rank of the Gram matrix.