What’s the version of your MOSS or WSS? Do you use Exchange server or just SMTP to send the email? Can you receive alert  when you add folder  or document  in the root directory  of this document library?

Your understanding of alert is right. Indeed there are basic steps to troubleshoot alert problem however first you need to find that if the alert can work  well on other document libraries. Please follow the steps below to test:
1. Create a new document library  and create a folder called “root folder” and set alert. Ensure that you receive the alert email after you set alerts.
2. Create a sub folder called “sub folder” under the root folder and wait for a while to see if you can get the email to notify this sub folder is created.
3. After you get the email, upload a test document in this subfolder to see if you can get email after a while.
If you can finish the steps above successfully, I think there is no problem with your alert. You need to check if the account which receives the email has at least read permission on the item in this document library.

If you cannot receive email at any steps above, you need to troubleshoot your alert referring to the article below:
msmvps.com/.../...hoot-alerts-in-wss-3-0-moss.aspx