Thermoacoustic-Stirling hybrid engines and feedback pulse tube refrigerators can utilize jet pumps to suppress streaming that would otherwise cause large heat leaks and reduced efficiency. It is desirable to use jet pumps to suppress streaming because they do not introduce moving parts such as bellows or membranes. In most cases, this form of streaming suppression works reliably. However, in some cases, the streaming suppression has been found to be unstable. Using a simple model of the acoustics in the regenerators and jet pumps of these devices, a stability criterion is derived that predicts when jet pumps can reliably suppress streaming.