High-entropy alloys (HEAs) are new alloys that contain five or more elements in roughly-equal proportion. We present new experiments and theory on the deformation behavior of HEAs under slow stretching (straining), and observe differences, compared to conventional alloys with fewer elements. For a specific range of temperatures and strain-rates, HEAs deform in a jerky way, with sudden slips that make it difficult to precisely control the deformation. An analytic model explains these slips as avalanches of slipping weak spots and predicts the observed slip statistics, stress-strain curves, and their dependence on temperature, strain-rate, and material composition. The ratio of the weak spots' healing rate to the strain-rate is the main tuning parameter, reminiscent of the Portevin-LeChatellier effect and time-temperature superposition in polymers. Our model predictions agree with the experimental results. The proposed widely-applicable deformation mechanism is useful for deformation control and alloy design.