Error message "Realms (429):" when trying to open Minecraft Realms

When pressing the "Minecraft Realms" button to bring up the list of realms I have joined, I get the following error message with no other details:

"Realms (429):"

Everything was working just fine less than 30 minutes ago with no new changes or configurations made since then.

Restarting Minecraft does not fix the issue.