The solubility and compactness of proteins is investigated within the framework of models amenable to an exact numerical study through exhaustive enumeration. We study how the average inter-amino acid interaction potential affects the properties of both isolated and interacting proteins. In a concentrated solution, depending on the value of the average potential, individual proteins may remain stable in the isolated native structure (soluble case), may aggregate preserving their geometrical shape (nonsoluble case) or aggregate changing their geometrical shape (prionlike behavior). The number of sequences that have compact native states and are soluble is maximal at a fine-tuned average interaction potential and of the same order of the corresponding number of nonsoluble prionlike proteins. The viable protein sequences selected by such a fine-tuned potential are found to have an amino acid composition similar to naturally occurring proteins.