Good points to remember from Navynurse2be and Curmudgeon. Anyone with internet access could be looking at this discussion, American or otherwise. However, nobody is trying to explain how to scam the folks who do background checks. To answer the original question...
As far as checking medical backgrounds - the physicals you get in the Navy will indicate many things (as an example, a dentist can tell if you are a smoker). For Secret clearance, however, I don't think that they are going to spend a whole lot of time and money looking for some nitpicky thing that happened umpteen years ago unless there is some indication that you are trying to hide it. They may talk to friends, family, or former coworkers about you, and if one of these people says something that interests them, they may follow up on it.
If a person has something completely diqualifying in their background (an OD on heroin or such) is would be a REALLY bad idea to try and cover it up. A good physical might uncover indications of that kind of thing, triggering a more thorough examination. For those with such things in their past, don't even apply for a job in the military. Also, they may, on a random basis, do an extremely thorough check. They could find out from former employers what health insurance a person had, and they could find out from public records if a person was treated without insurance for a condition that resulted from criminal activity.
Best answer - if you've got something like that in your background, look for another job. Otherwise, focus on your new job and let the investigators do theirs...if you are not sure if something in your past would cause you to fail a background check, keep it to yourself unless specifically asked. Then, as always, tell the truth. The condition may not be disqualifying in itself, but lying about it certainly would be.
-A
Ummm...yeah