When you specify the MessagingServiceSid parameter in the API request, Twilio will first return an ‘accepted’ status, and then determine the optimal 'From' phone number. Once a From number is assigned, the message will be moved to the "queued" status, waiting to be processed and handed off to our carrier partner for delivery.

You will see messages sitting in the "accepted" state if all the numbers in the pool are being used. To resolve this issue, simply add more Twilio phone numbers to the Messaging Service to increase capacity.

You can add numbers to the service by following the guide here.