The cluster of galaxies A2256 was observed by the PCA and HEXTE experiments aboard the Rossi X-Ray Timing Explorer satellite during the period 2001 July - 2002 January, for a total of similar to 343 and similar to 88 ks, respectively. Most of the emission is thermal, but the data analysis yields evidence for two components in the spectrum. On the basis of statistical likelihood alone, the secondary component can be either thermal or power law. Inclusion in the analysis of data from ASCA measurements leads to a more definite need for a second component. Joint analysis of the combined RXTE+ ASCA data sets yields kT(1) = 7.9(-0.2)(+0.5) keV and kT(2) = 1.5(-0.4)(+1.0) keV, when the second component is also thermal, and kT = 7.7(-0.4)(+0.3) keV and alpha = 2.2(-0.3)(+0.9) if the second component is fitted by a power law with (photon) index alpha; all errors are at 90% confidence. Given the observed extended regions of radio emission in A2256, it is reasonable to interpret the deduced power-law secondary emission as due to Compton scattering of the radio-producing relativistic electrons by the cosmic microwave background radiation. If so, then the effective, mean volume-averaged value of the magnetic field in the central 1degrees region of the cluster - which contains both the "halo'' and "relic'' radio sources - is B similar to 0.2(-0.1)(+1.0) muG.