There could be other things to cause it other than a badly design ground plane. Its sounds like the typical problem when you use a non isolated power supply. Which one way to get rid of it is a .1uf capacitor with a 100 ohm resistor in series. But without inspecting the circuit, if you are trying to use a RS232 driver with a single supply it does this. They require +5 and -5 supplies.The noisy device has one of those RFID 13.56MHz modules with onboard etched antennas. I put a lot of caps to keep the power as clean as possible, but I never realized the ground plane was so noisy until I tried to send serial data to another device. By only connecting the two ground planes it drives the second device insane.