If you are an internal user, with either the "Administrator", "Manager" or "Claim Handler" role, you can potentially delete claims.
Administrators can always delete claims, but for Managers and Claim Handlers the permission is optional. For Managers the setting is on by default, but for Claim Handlers it needs to be enabled.
Administrators can enable the "Can Delete Claims?" permission for other users from the Users sections under Settings. See User Permissions for more details.