What does a PhD application rejection mean for other applications? If you applied to several PhD projects, all in the same "level" of university prestige.

Could a refusal from one of these applications be interpreted as a inadequacy for the rest?. Could it mean the applicant should start aiming for one "level" lower?
Virtually everyone gets rejected from at least one of the places they apply to. I have never heard of anyone who got through the process without getting rejected somewhere. I even got rejected from several places that were lower ranked than the ones I did get in to!

So no, one single rejection is not a sign that you are applying to the wrong level of universities. In fact, you should probably expect more rejections. Depending on the subject there can be 5-10 qualified applicants for each spot, so lots of good candidates are rejected every year. If you discussed what universities to apply to with the professors writing your recommendation letters, you are probably applying to universities of the right "level". The sheer number of applications means that you will get some rejections anyway.

With that said: if your are really set on graduate school, it is a good idea to apply to a wide range of schools on different levels.