We make a critical reappraisal ofùnbounded-from-below' (UFB) constraints in the MSSM and R-parity violating models. We explain why thètraditional' UFB bounds are neither necessary nor suucient and propose, instead, a suucient condition which ensures that there are no local minima along the at directions. This conservative (but meaningful) condition divides the parameter space into regions which are allowed , regardless of cosmology, and regions in which cosmology is expected to play a major role. We study both conditions at low tan and obtain analytic approximations to the UFB bounds for all low tan (< 15). Finally we show that R-parity violation just below current experimental limits avoids UFB problems by lifting the dangerous at directions.