Cannot connect to Realms, "Realms (429) error"

Upon clicking the button labeled "Minecraft Realms" on the title screen, I am presented with a screen that simply reads "Minecraft (429)", with nothing else but a confirmation button. Upon pressing the button, I just get taken back to the title screen. The times that this happens are random, it'll break and then an hour later it works fine. I've made no modifications to my game recently, and this comes up out of seemingly nowhere. I've done the usual stuff to fix issues, such as restarting Minecraft and my PC, as well as turning off automatic Java updates to see if that does anything (it does nothing).