Seasonal prediction systems based on Earth System Models exhibit a lower proportion of predictable signal to unpredictable noise than the actual world. This puzzling phenomena has been widely referred to as the signal-to-noise paradox (SNP). Here, we investigate the SNP in a conceptual framework of a seasonal prediction system based on the Lorenz, 1963 Model (L63). We show that the SNP is not apparent in L63, if the uncertainty assumed for the initialization of the ensemble is equal to the uncertainty in the starting conditions. However, if the uncertainty in the initialization overestimates the uncertainty in the starting conditions, the SNP is apparent. In these experiments the metric used to quantify the SNP also shows a clear lead-time dependency on sub-seasonal timescales. We therefore, formulate the alternative hypothesis to previous studies that the SNP could also be related to the magnitude of the initial ensemble spread.