certainly they could. Islands (many of them) are the surface of underwater mountains...made by volcanic activity. Wind and wave erode the surface....they go away?
Also..what happens when the inhabitants pump out all of the fresh water underground? What holds the surface up when they do?
Also...suppose the islands are on a tectonic plate that is subducting under the plate next to it?
Also..suppose the island in question is on the backside of a plate that is undergoing isotatic rebound?
Also...what about the wobble of the earth's axis...it changes over time. How does that effect the tides?
Also...well that's enough for now.