Carbon nanotube (CNT) fibers are a promising material for wearable electronics and biomedical applications due to their combined flexibility and electrical conductivity. To engineer the bending properties for such applications requires understanding how the bending stiffness of CNT fibers scales with CNT length and fiber diameter. We measure bending stiffness with a cantilever setup interpreted within Euler Elastica theory. We find that the bending stiffness scales with a power law of 1.9 for the fiber diameter and 1.6 for the CNT length. The diameter scaling exponent for fiber diameter agrees with results from earlier experiments and theory for microscopic CNT bundles. We develop a simple model which predicts the experimentally observed scaling exponents within statistical significance.