Neutron production double-differential cross sections from thin C and Nb targets bombarded by 290 MeV/u $^{136}$Xe ions were measured by using the HIMAC (Heavy Ion Medical Accelerator in Chiba) facility of National Institute of Radiological Sciences, Japan. We used the inverse time-of-flight method using NE213 liquid organic scintillators. NE213 liquid organic scintillators were at the angles of 15°, 30°, 45°, 60°, 75°, 90°. For measuring the background, iron bars were set for each flight path. In order to shield scattering particles from a beam dump, iron and concrete blocks were placed between the detectors and the beam dump. The experimental results were compared with Monte Carlo codes PHITS and GEANT4. Monte Carlo results generally agree with the measured ones within a factor of two.