We have investigated this with the manufacturer, and there are some reports about a firmware update that causes these error messages on some printers only when the printer is powered off, or if the lid is opened for a replacement cartridge. As long as you override the error message by pressing and holding the green "check" button, and as long as you leave the printer on, there should be no error messages after that until it comes time to replace the cartridge. Unfortunately this is the only workaround.
HP Laserjet 4200, 4250, 4300, and 4350