When I've worn rotors down they get noisy easier and get the "turkey warble" sound. Also, a lip forms on the outside diameter which makes it harder to get the wheel back on. I only have discs on my mtb but I have worn rotors way past the minimum to the point where I worry about the front one bending under heavy load.

I've also measured the brake track on a rim brake and found it to be .025 inch lower than when new.

I use a machinist's micrometer to measure thickness but just out of curiosity. It's not super accurate anyway since the micrometer can't reach the low spots on the surface of the rotor. Automotive rotor micrometers and brake drum micrometers have pointed tips for this reason.