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I would assume that dropping flow rates or a drop in chamber pressure would cause the engine to stop thrusting, but I am unaware as to what specifically these conditions might be.

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  • $\begingroup$ if the exhaust flow fails to reach the speed of sound in the engine throat then it will slow, instead of speed up as it exits the nozzle, resulting in a loss of thrust. $\endgroup$ – JCRM Mar 26 '18 at 23:21
  • $\begingroup$ Another condition would be if external pressure is higher than chamber pressure such as deep underwater or in a high pressure environment such as on a gas giant. $\endgroup$ – Dragongeek Mar 27 '18 at 21:09
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When thinking about what could happen, a fruitful method of approach can be considering what has happened.

This chart from the early days of SSME testing shows some of the ways things can go wrong.

  • Turbopump failures resulting in loss of flow and Other Bad Things

  • Burn-throughs of combustion machinery

  • Ruptures of piping
  • Valve failures

enter image description here

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