Many amyloid proteins form metastable soluble aggregates (or protofibrils, or protein nanoparticles, with characteristic sizes from approximately 10 to a few hundred nm). These can coexist with protein monomers and amyloid precipitates. These soluble aggregates are key determinants of the toxicity of these proteins. It is therefore imperative to understand the physical basis underlying their stability. Simple nucleation theory, typically applied to explain the kinetics of amyloid precipitation, fails to predict such intermediate stable states. We examine stable nanoparticles formed by the Alzheimer's amyloid-beta peptide (40 and 42 residues), and by the protein barstar. These molecules have different hydrophobicities, and therefore have different short-range attractive interactions between the molecules. We also vary the pH and the ionic strength of the solution to tune the long-range electrostatic repulsion between them. In all the cases, we find that increased long-range repulsion results in smaller stable nanoparticles, whereas increased hydrophobicity produces the opposite result. Our results agree with a charged-colloid type of model for these particles, which asserts that growth-arrested colloid particles can result from a competition between short-range attraction and long-range repulsion. The nanoparticle size varies superlinearly with the ionic strength, possibly indicating a transition from an isotropic to a linear mode of growth. Our results provide a framework for understanding the stability and growth of toxic amyloid nanoparticles, and provide cues for designing effective destabilizing agents.