Several aggregation-prone RNA-binding proteins, including FUS, EWS, TAF15, hnRNP A1, hnRNP A2, and TDP-43, are mutated in neurodegenerative diseases. The nuclear-cytoplasmic distribution of these proteins is controlled by proteins in the karyopherin family of nuclear transport factors (Kaps). Recent studies have shown that Kaps not only transport these proteins but also inhibit their self-association/aggregation, acting as molecular chaperones. This chaperone activity is impaired for disease-causing mutants of the RNA-binding proteins. Here, we review physical data on the mechanisms of self-association of several disease-associated RNA-binding proteins, through liquid-liquid phase separation and amyloid fiber formation. In each case, we relate these data to biophysical, biochemical, and cell biological data on the inhibition of self-association by Kaps. Our analyses suggest that Kaps may be effective chaperones because they contain large surfaces with diverse physical properties that enable them to engage multiple different regions of their cargo proteins, blocking self-association.