The length of nanotubes is a critical structural parameter for the design and manufacture of nanotube-based material systems and devices. High-precision length control of nanotubes by means of mechanical cutting using a scriber has not materialized due to the lack of the knowledge of the appropriate cutting conditions and the tube failure mechanism. In this paper, we present a quantitative nanomechanical study of the cutting of individual boron nitride nanotubes (BNNTs) using atomic force microscopy (AFM) probes. In our nanotube cutting measurements, a nanotube standing still on a flat substrate was laterally scribed by an AFM tip. The tip-tube collision force deformed the tube, and eventually fractured the tube at the collision site by increasing the cutting load. The mechanical response of nanotubes during the tip-tube collision process and the roles of the scribing velocity and the frictional interaction on the tip-tube collision contact in cutting nanotubes were quantitatively investigated by cutting double-walled BNNTs of 2.26-4.28 nm in outer diameter. The fracture strength of BNNTs was also quantified based on the measured collision forces and their structural configurations using contact mechanics theories. Our analysis reports fracture strengths of 9.1-15.5 GPa for the tested BNNTs. The nanomechanical study presented in this paper demonstrates that the AFM-based nanomechanical cutting technique not only enables effective control of the length of nanotubes with high precision, but is also promising as a new nanomechanical testing technique for characterizing the mechanical properties of tubular nanostructures.