Route 53 - Received message - We can't finish registering your domain

When you encounter an issue with domain registration in Amazon Route 53 and receive a message stating "We can't finish registering your domain," it's a common problem that can occur for various reasons. These reasons may include issues with the domain name itself, problems with the registration information provided, or technical difficulties on AWS's end.

After opening a support case for a domain registration issue, as you have done, the response time can vary. AWS does not provide a specific timeframe for responding to these types of support tickets. However, domain registration issues are generally handled under the Account and Billing support category, which is available to all AWS customers regardless of their support plan level.

While waiting for a response, it's important to note that AWS Support aims to address these issues as quickly as possible. The complexity of your specific case may influence the response time. In the meantime, you can try the following:

1. Double-check that you've provided all necessary information in your support case.
2. Ensure that the AWS account you used to open the support case is the same one you used for the domain registration attempt.
3. If you haven't already, you could try the registration process again, carefully reviewing all the information you're entering to ensure accuracy.

If you don't receive a response within a reasonable timeframe (e.g., a few business days), you may want to follow up on your existing case or consider opening a new one with additional details about your registration attempt.

Remember, AWS Support is the most authoritative source for resolving these types of issues, as they have direct access to your account information and can provide specific guidance based on your situation.
Sources
Amazon Route 53 | AWS re:Post
Fail to register a new domain | AWS re:Post
About domain registration on Route 53 | AWS re:Post