We have observed an anomalous monotonically increasing longitudinal field relaxation rate in LiY_(1-x)Ho_xF_4 for applied fields above 1000 G (along the c-axis) and at a temperature of 16 K for samples with x = 0.01 and 0.10. This highly unusual behavior, observed to persist up to 6000 G at the PSI facility, may be related to enhanced spin relaxation at an avoided level crossing near 1.8 T in the Ho electronuclear energy level diagram; other possible explanations include unusual field-dependent spin-phonon or spin-orbit coupling or changes in the thermal populations of the Zeeman-split energy levels. We request 2 days of beamtime in the HiFi spectrometer to monitor the relaxation rate to fields greater than 2 T and in a range of temperatures to determine the origin of this highly unsual behavior.