Could a lipo/18650 overheat during discharge?
Your cell will heat by dissipating I^2R power through its' internal resistance. Since the resistance is inside the cell and your protection circuit is measuring the voltage at the terminals, the circuit will always see this as a voltage drop. So, at the very least, there is a sign that all is not well inside.

As for your question, it depends on a lot of things, perhaps most importantly on whether your protection circuit also has a temperature sensor. :) For example, if you wrapped your cell in lots of insulation and then used it at its' current limit, there's a decent chance it would overheat, regardless of its' state-of-health. 

Bottom line? A well-constructed (and designed) solution used as specified in said design should not fail, regardless of age. Since you mentioned cheap-ish in your question, I'd keep some fire safety equipment handy. :)