My employment ended with my company on 7/25/08, and my insurance ends on 7/31/08. I do not start my new job until 8/4/08. So I have a 4 day gap in coverage. Do I need to sign up for COBRA in advance? Or can I sign up retroactively, if something happens during this 4 day period?

You would have to have signed up prior to needing the coverage.

You can't wait until you are sick, to get the insurance to cover it.

Well actually after you left your employment, the insurance company will mail you the cobra paper work, you can't even sign up for it, till they mail it to you, In theory you are suppose to sign up as soon as you leave your employment ( also are you 100 percent sure you insurance continues to end of month, it is very common for it to end the day your employment ends, And cobra to start then, But call the cobra is nothing more than your current insurance plan, that you pay a lot for though your insurance company, so call the toll free number on the insurance card and ask them when you are going to get your cobra paperwork, and what ahppens if you get sick before you get the paper work.

COBRA regulations allow you 60 days to decide if you want to sign up for COBRA as long as Federal COBRA regulations apply to you. You need to have been employed by a company with 20 or more employees. Or live in a state that has their own "COBRA" regs in place for smaller companies.
It's pretty common to wait on the COBRA decision until you have coverage at a new job. If you have not needed it you move on to the new coverage. If you do need to sign up for it though, you will need to pay retroactive back to the day you were eligible. In your case 8/1.
So the simple answer is yes, with COBRA, you can wait to see if you need it.