Many studies have pointed out fractal and multifractal properties of photospheric magnetic fields, but placing the various approaches into context has proved difficult. Although fractal quantities are defined mathematically in the asymptotic limit of infinite resolution, real data cannot approach this limit. Instead, one must compute fractal dimensions or multifractal spectra within a limited range at finite scales. The consequent effects of this are explored by calculation of fractal quantities in finite images generated from analytically known measures and also from solar data. We find that theorems relating asymptotic quantities need not hold for their finite counterparts, that different definitions of fractal dimension that merge asymptotically give different values at finite scales, and that apparently elementary calculations of dimensions of simple fractals can lead to incorrect results. We examine the limits of accuracy of multifractal spectra from finite data and point out that a recent criticism of one approach to such problems is incorrect.