The Laser Interferometer Space Antenna (LISA) will search for a stochastic gravitational-wave (GW) background at millihertz frequencies, from both astrophysical and cosmological sources, and thereby open a new chapter in GW astronomy. In the literature, LISA’s sensitivity to prospective GW background (GWB) signals is often quantified in terms of an expected signal-to-noise ratio (SNR) assuming perfect knowledge of the detector noise. The commonly employed expression for the SNR is, however, valid only in the limit of a weak GWB signal, which renders a large number of SNR values reported in the literature inaccurate. In this paper, we address this issue by leveraging an expression for the expected optimal SNR of a LISA auto-correlation measurement that is valid at arbitrary signal strength. Based on this generalized expression, we conclude that LISA data worth an observing time of T_obs across the frequency band from f_min to f_max will never yield an SNR in excess of SNR_max = , which evaluates to SNR_max ^4 for typical mission parameters. We illustrate our findings in terms of generalized power-law-integrated (PLI) sensitivity curves at different SNR levels, i.e., LISA SNR contour lines in plots of the GW energy-density power spectrum. In contrast to earlier work on PLI sensitivity curves, we find that the LISA SNR contours are bounded from above, approximately by the LISA strain noise curve multiplied by a factor of Euler’s number e. Our numerical results for the LISA SNR contours, including our new SNR cutoff, are available on Zenodo [1], readily allowing for the evaluation of the expected SNR for a LISA auto-correlation measurement.