How much can you bend a flex PCB?
As with most engineering questions, the answer is "it depends". 

You can purchase flex circuits optimized for frequent bending (dynamic) applications, and you should minimize the copper thickness, and ideally use just a single layer, to make the PCB as robust as possible. Tight radius bending is bad, but how bad depends on the flex construction (number of layers, substrate), and thickness and type of copper you specify from the maker. If you need multiple layers, consider using more than one single or two layer flex. 

Repeated bending with a tight radius (a small multiple of the substrate thickess) should be avoided.