For the evaluation of the detection and measurement limits for gamma-spectrometry measurements formulas were elaborated assuming the presence of the background component, which correlates with the analytical peak in the sense of: (1) linear relation between the expectation value of the peak and background counts, and (2) statistical correlation between peak and background counts. It is shown that neglecting the correlated background component can lead to a significant overestimation of the detection and measurement limits. The problem of evaluating of the correlated background component by applying experimental and computational approaches is also considered.