A common problem in experimental data analysis is to locate the position of a signal to an accuracy which is substantially less than the actual signal width. By applying maximum likelihood estimation to this problem, this paper derives theoretical limits on the ability to locate signal position. The limiting error in position measurement is shown to be a simple function of the instrument resolution, the density of sample points, and the signal-to-noise ratio of the data. An interesting conclusion is that position information on a much finer scale than the minimum instrument sampling interval is contained in data of modest signal-to-noise ratio. The common procedure of excluding the portion of the data lying below an amplitude threshold to guard against background fluctuations is incorporated in the maximum likelihood analysis. It is shown that selection of the optimum amplitude threshold level depends on the type of noise present in the data, and can be an important factor in position accuracy. The analytical results exhibit close agreement with Monte Carlo simulations of position accuracy in the presence of noise.