We are definitely not hitting Rate limit on our small Bitbucket server.
Other users have reported issue: https://github.com/Eernie/bitbucket-webhooks-plugin/issues/72

Latest Bitbucket branch source plugin
So we have no builds until you hit "Scan" then events start working for a little while and then you have to scan again.

See attachments

Here I search for remove... Notice the large gap in between removed and created/updated events.

Here I hit "Scan" as indicated by the green. Notice the one hour time gap