@Don Costanzo 429 error code indicates that the rate of requests sent to the API exceeds the safe threshold limit. In most cases, the Tranactions per second(TPS) value is hit and you will see this message. Depending on the AI service and API you are using, there could be an option to increase the TPS limits from quota limits page of the service or request a higher TPS by raising a support case from Azure portal.
If you have created your service or resource with free tier, try changing that to a higher or standard tier and then retry the scenario. In most cases, this is the cause of the message.
Also, please note that the services are designer to scale up as per demand and if you see the same error while increasing the workload try to avoid sharp increase in workload, some of the best practices for these scenarios are also documented in the respective service limits documentation. For example, for speech services you can look up the guidance here.