We discuss the effect of large positive correlations in the combinations of several measurements of a single physical quantity using the Best Linear Unbiased Estimate (BLUE) method. We suggest a new approach for comparing the relative weights of the different measurements in their contributions to the combined knowledge about the unknown parameter, using the well-established concept of Fisher information. We argue, in particular, that one contribution to information comes from the collective interplay of the measurements through their correlations and that this contribution cannot be attributed to any of the individual measurements alone. We show that negative coefficients in the BLUE weighted average invariably indicate the presence of a regime of high correlations, where the effect of further increasing some of these correlations is that of reducing the error on the combined estimate. In these regimes, we stress that the correlations provided as input to BLUE combinations need to be assessed with extreme care. In situations where the precise evaluation of these correlations is impractical, or even impossible, we provide tools to help experimental physicists perform more conservative combinations.