The mechanical properties of individual multiwall boron nitride nanotubes (MWBNNTs) synthesized by a growth-vapor-trapping chemical vapor deposition method are investigated by a three-point bending technique via atomic force microscopy. Multiple locations on suspended tubes are probed in order to determine the boundary conditions of the supported tube ends. The bending moduli (E-B) calculated for 20 tubes with diameters ranging from 18 to 58 nm confirm the exceptional mechanical properties of MWBNNTs, with an average E-B of 760 +/- 30 GPa. For the first time, the bending moduli of MWBNNTs are observed to increase with decreasing diameter, ranging from 100 +/- 20 GPa to as high as 1800 +/- 300 GPa. This diameter dependence is evaluated by Timoshenko beam theory. The Young's modulus and shear modulus were determined to be 1800 +/- 300 and 7 +/- 1 GPa, respectively, for a trimmed data set of 16 tubes. The low shear modulus of MWBNNTs is the reason for the detected diameter-dependent bending modulus and is likely due to the presence of interwall shearing between the crystalline and faceted helical nanotube structures of MWBNNTs.