Among the faculty members I've known (at US research universities), there's a widespread feeling that recommendation letters should be completely confidential unless the writer chooses otherwise, and that viewing them using FERPA is unethically taking advantage of a legal loophole. Almost all the students check the waiver box, and if someone doesn't, then the letter writer is more likely to assume it was by accident than on purpose.
In this context, my advice is:
You should always check the box. If your recommenders believe you may look at the letters later, they will probably write weaker, vaguer letters. (For example, the most compelling letters often involve comparisons with other students, which may be omitted if you'll be reading the letter.)
If you are not checking the box on purpose, you should say so explicitly. If you do this silently, people may assume it was an accident, and then if they learn later that you viewed the letters, they will be more offended than if you had announced this plan in advance. (And they may actually find out, since the staff who handle your request may find it troubling and leak the information even if they aren't supposed to.)
You would learn less than you might expect from looking at the letters. It's remarkably hard to judge letters out of context, without having seen other letters from the same people, and it's not likely you'll discover a clear reason for your rejection. If anything, it might mislead you: you might decide that Professor X was damaging your chances by being insufficiently enthusiastic, without realizing that Professor X is never enthusiastic and in context this letter was viewed as very positive.
I think the fundamental worry many students have is of a terrible letter, a single letter that ruins what would otherwise have been a successful application. This can happen, but I see an example only once every few years. And even in those cases, it often looks like it should have been predictable to the applicant. (For example, if you have had difficulties with someone in the past but things seem better now, don't ask for a letter without a serious discussion of how they think things stand now.) So I wouldn't worry too much: the chances you could dramatically improve your application by substituting a letter are small.