Protein structure determination by NMR has predominantly relied on simulated annealing-based conformational search for a converged fold using primarily distance constraints, including constraints derived from nuclear Overhauser effects, paramagnetic relaxation enhancement, and cysteine crosslinkings. Although there is no guarantee that the converged fold represents the global minimum of the conformational space, it is generally accepted that good convergence is synonymous to the global minimum. Here, we show such a criterion breaks down in the presence of large numbers of ambiguous constraints from NMR experiments on homo-oligomeric protein complexes. A systematic evaluation of the conformational solutions that satisfy the NMR constraints of a trimeric membrane protein, DAGK, reveals 9 distinct folds, including the reported NMR and crystal structures. This result highlights the fundamental limitation of global fold determination for homo-oligomeric proteins using ambiguous distance constraints and provides a systematic solution for exhaustive enumeration of all satisfying solutions.