Superconducting-insulating-superconducting (SIS) tunnel junctions coupled to superconducting absorbers may be used as high-resolution, high-efficiency X-ray spectrometers. Until recently, the X-ray-induced current pulse from such devices has been measured using FET-based negative-feedback charge or current amplifiers. The limited bandwidth and feed-back nature of these amplifiers have made it difficult to deduce the true shape of the X-ray induced current pulse. Recently, we have begun to use high-bandwidth amplifiers based on Superconducting Quantum Interference Devices (SQUIDS) to measure the current pulses from our tunnel junction X-ray spectrometers. We have measured pulses from devices with niobium X-ray absorbing layers coupled to aluminum layers that serve as quasiparticle traps. We present here a study of pulse shape as a function of bias voltage. In general, the X-ray induced pulses increase in amplitude and become longer as we increase the bias voltage. We found that it is possible to differentiate pulses produced by X-ray absorption in the top niobium film from those produced in the bottom niobium film by measuring the rise time of the current pulses. This allows us to produce a high resolution spectrum using only pulses produced in the bottom niobium film. The measured energy resolution of this spectrum is 29 eV FWHM at 5.89 keV, about 5 times better than that obtainable using semiconductor ionization detectors.