Enhanced protein aggregation and/or impaired clearance of aggregates can lead to neurodegenerative disorders such as Alzheimer's Disease, Huntington's Disease, and prion diseases. Therefore, many protein quality control factors specialize in recognizing and degrading aggregation-prone proteins. Prions, which generally result from self-propagating protein aggregates, must therefore evade or outcompete these quality control systems in order to form and propagate in a cellular context. We developed a genetic screen in yeast that allowed us to explore the sequence features that promote degradation versus aggregation of a model glutamine/asparagine (Q/N)-rich prion domain from the yeast prion protein, Sup35, and two model glycine (G)-rich prion-like domains from the human proteins hnRNPA1 and hnRNPA2. Unexpectedly, we found that aggregation propensity and degradation propensity could be uncoupled in multiple ways. First, only a subset of classically aggregation-promoting amino acids elicited a strong degradation response in the G-rich prion-like domains. Specifically, large aliphatic residues enhanced degradation of the prion-like domains, whereas aromatic residues promoted prion aggregation without enhancing degradation. Second, the degradation-promoting effect of aliphatic residues was suppressed in the context of the Q/N-rich prion domain, and instead led to a dose-dependent increase in the frequency of spontaneous prion formation. Degradation suppression correlated with Q/N content of the surrounding prion domain, potentially indicating an underappreciated activity for these residues in yeast prion domains. Collectively, these results provide key insights into how certain aggregation-prone proteins may evade protein quality control degradation systems.