Extended x-ray absorption fine structure (EXAFS) at the Cd K edge and diffraction patterns have been measured on CdTe as a function of pressure from 100 kPa (1 bar) to 5 GPa using a cell with nano-polycrystalline diamond anvils and an x-ray focussing scanning spectrometer. Three phases-zincblende (ZB), mixed cinnabar-ZB and rocksalt (RS)-are well distinguished in different pressure intervals. The bond compressibility measured by EXAFS in the ZB phase is slightly smaller than the one measured by diffraction and decreases significantly faster when the pressure increases; the difference is attributed to the effect of relative vibrations perpendicular to the Cd-Te bond. The parallel mean square relative displacement (MSRD) decreases, the perpendicular MSRD increases when the pressure increases, leading to an increasing anisotropy of relative atomic vibrations. A constant-temperature bond Grüneisen parameter (GP) has been evaluated for the ZB phase and compared with the constant-pressure bond GP measured in a previous experiment; an attempt is made to connect the bond GPs measured by EXAFS and the more familiar thermodynamic GP and mode GPs; the comparisons suggest the inadequacy of the quasi-harmonic approximation to deal with the local vibrational properties sampled by EXAFS.