ABSTRACT: We investigate the quantum mechanical bound to how precisely we can determine a phase shift given only a constraint on the mean total number of photons available. By considering how quickly one can gain information from data analysis, we derive the sensitivity achievable (in principle) for measurements involving even highly non-Gaussian noise. Using these results we calculate the sensitivity of several recent proposals for precision phase measurement, and show that no proposal to date beats the sensitivity believed achievable by squeezed state interferometry.