The ability of the atomic force microscope to measure forces with subnanonewton sensitivity at nanometerscale lateral resolutions has led to its use in the mechanical characterization of nanomaterials. Recent studies have shown that the atomic force microscope can be used to measure the elastic moduli of suspended fibers by performing a nanoscale three-point bending test, in which the center of the fiber is deflected by a known force. We extend this technique by modeling the deflection measured at several points along a suspended fiber, allowing us to obtain more accurate data, as well as to justify the mechanical model used. As a demonstration, we have measured a value of 78 +/- 17 GPa for Young's modulus of bacterial cellulose fibers with diameters ranging from 35 to 90 nm. This value is considerably higher than previous estimates, obtained by less direct means, of the mechanical strength of individual cellulose fibers.