In set membership estimation, conditional problems arise when the estimate must belong to a given set of specified structure. Central and interpolatory projection algorithms provide conditional estimates that are suboptimal in terms of the worst-case estimation error. In order to precisely evaluate the suboptimality level of these estimators, tight upper bounds on the estimation error must be computed as a function of the conditional radius of information, which represents the minimum achievable error. In this paper, tight bounds are derived in the V ∞ and V 1 cases, for a general setting which allows to consider any compact set of feasible problem elements and linearly parametrized estimates.