It is often the case that economic mechanisms can have hidden characteristics with unexpected consequences in practice. This is especially true when real-world budgets come into play. For instance, the infamous German and British 3G spectrum auctions that generated revenue so successfully that cellular service providers delayed the rollout of 3G networks for lack of funds. We contribute a piece of the puzzle by characterizing the space of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto-optimal combinatorial auctions where efficiency does not hold. We examine a model with two players and k nonidentical items (2k outcomes), multidimensional types, private values, nonnegative prices, and quasilinear preferences for the players with one relaxation - one of the players is subject to a publicly-known budget constraint. We show that if it is publicly known that the valuation for the largest bundle is more than the budget for at least one of the players then the following are true. (a) VCG does not fulfill the basic properties of deterministic, dominant-strategy incentive compatible, individual rationality and Pareto optimality when the all-item bundle is not arbitrarily allocated. (b) Of the dictatorial solutions, only a single family of non-trivial dictatorial mechanisms fulfills the above basic properties. (c) Weakening the public knowledge assumption results in no VCG nor dictatorship mechanisms that fulfill the properties. Our characterization of the non-efficient space for deterministic budget-constrained combinatorial auctions is similar in spirit to that of  for Bayesian single-item constrained efficiency auctions.