Single source localization from low-pass filtered measurements is ubiquitous in optics, wireless communications and sound processing. We analyse the performance of the maximum likelihood estimator (MLE) in this context with additive white Gaussian noise. We derive necessary conditions and sufficient conditions on the maximum admissible noise level to reach a given precision with high probability. The two conditions match closely, with a discrepancy related to the conditioning of a noiseless cost function. They tightly surround the Cramér-Rao lower bound for low noise levels. However, they are significantly more precise for larger levels. An outcome is new optimization criteria for the design of point spread functions in single molecule microscopy.