In theory maybe, in practice no. UV damage is a bigger threat to ropes.
Climbers repeatedly tie knots (which temporarily reduce the strength of a rope) into the ends of a rope. Then they proceed to fall on that rope with often high amounts of force repeatedly. Their ropes don't break due to the strain and repeated knots.
Sailors will sail with nylon ropes in their rigging, at times these ropes might flog in the wind. The flogging ends don't break off because of repeated bending/flogging. Similarly loads secured to a trailer or vehicle will flog in the wind. The ends of these ropes don't snap off either.
At the end of the day a rope is like a woven product in that many smaller fibers work together to become a whole that is stronger than the parts. This lets individual fibers fail due to fatigue, excess load, abrasion, UV rays, etc. without really affecting the whole until a critical mass has been reached. In general I would expect a rope to withstand at least as many bends as your shirt, aka lots.
Something else to consider, in general the fibers in a rope won't be bent like the metal in a paper clip. The woven nature of the rope will allow the fibers to pull slightly in the weave so the pressure is exerted over a larger area than a similar bend in a paper clip. Also nylon fibers, I believe, have some elasticity. This allows them to stretch. Metals or rigid plastics are typically what you think of "fatiguing." That is because they don't have the same amount of elasticity. I believe this elasticity is what allows a rubber band to bend without fatigue.
