Studies in vitro show that globular proteins can experience the formation of native-like conformational states able to self-assemble with no need of transitions across the energy barrier for unfolding, and that such processes can lead eventually to the formation of amyloid-like species. Circumstantial evidence collected in vivo suggests that aggregation of native-like states can be a concrete possibility for living organisms and thus more relevant than previously thought. In this review we summarize the key observations collected on the "native-like aggregation" of the acylphosphatase from Sulfolobus solfataricus, a protein that has allowed the direct monitoring and analysis of native-like aggregates for its propensity to form rapidly native-like aggregates and their slow conversion into amyloid-like aggregates.