The discussions around using Postman for managing API calls highlight the challenges related to rate limits encountered by users. One comment points to the 'hug of death' issue, which often occurs under high traffic situations. Users reported running into rate limit errors when exceeding their organization’s token usage quota. In response, an alternative suggestion was made to try out the Claude client, which currently has a more restricted request allowance. Additionally, there's a call for greater flexibility allowing users to manage their API token settings to mitigate the issue.