Boron nitride nanotube (BNNT)/polyvinyl alcohol (PVA) composite fibers (<5 vol % BNNTs) were fabricated via electrospinning so that all BNNTs became aligned in the fiber casting direction. A several-fibers-thick ensemble of parallel-arranged contacting fibers made a single polymer sheet. Numerous sheets were then stacked in different ways with respect to the BNNT orientation (all fibers in adjacent sheets were either parallel or alternately rotated 90°) to make multilayer films that were finally hot-pressed. Various BNNT textures were reflected by the corresponding differences in the measured thermal conductivities of the resultant films due to anisotropy of thermal transport in the nanotubes. The highest values (0.54 W/mK) were obtained along the long axes of aligned BNNTs. Somewhat lower values (0.38 W/mK) were documented in films with alternately stacked fibers/tubes. The theoretical thermal conductivity values were estimated using the Nielsen’s model. These show good match with the experimental data. ...