The billionaire announced the "temporary measure" to address extreme levels of data scraping on the site.
Initial limits were quickly increased by Mr Musk at the weekend.
While many users reported no longer seeing limits on Sunday, some said a "rate limit exceeded" notification had returned on Monday.
Mr Musk - who took over Twitter in October 2022 had said previously that he was not happy about artificial intelligence (AI) firms using Twitter's data to train their large language models.
Changes to the platform at the weekend saw it impose an initial 600-tweet limit for unverified Twitter users who are not paying for a subscription to the platform, but Mr Musk said this had increased to 1,000 on Saturday evening.
He has not yet provided an update on whether the limits will remain in place.