There are tonnes of reasons people can get better, ranging from the trivial to the remarkable. The point about a "cure" is that it has to be relatively general, ie useful for more than just one person, so one case isn't really enough to prove anything. On the other hand, it can't be rejected conclusively either.
Still, someone wanting to establish that a miracle cure is useful first has to ask, and reject, the alternative explanations of (among others): regression to the mean (ie, patient just getting better anyway, or looking at the patient at the lowest and highest parts of a cycle of illness); something else responsible other than what you are claiming; misdiagnosis in the first place; an unusual reaction that isn't replicated with other people; placebo.
A few others I might have missed. But just as anyone selling this stuff ought to try and rule it out, you can't really prove it's any one of those either without a possibly expensive set of exhaustive trials that you might have no motivation to perform. Because, after all, miracle cures are usually exactly that.