DVCS Accounts show sync errors for GitHub repositories in Jira Software
The DVCS plugin uses the eclipse git plugin to access GitHub. The git plugin from eclipse keeps track of the rate limit for a connection, but doesn't take any action if/when it reaches the limit. If the next synchronization starts before the rate limit is reset, the synchronization will fail again. When the synchronization reaches the rate limit, the currently processed message will fail and attempted to be applied in the next synchronization.
For JIRA Server, you can try to add the dvcs.connector.pr-synchronization.disabled dark feature to disable pull request synchronization as a way to reduce the initial sync load on GitHub. Then, re-enable that feature later when load is back to normal.
For JIRA Cloud, you can request that support verify the API limit is being reached by checking your JIRA logs or if you already suspect that as being the root cause, keep watching - DCON-211Getting issue details... STATUS for updates.