There are many examples of scribes attaining high positions in Ancient Egypt, they often make sure to leave evidence of how big a deal they were. Ancient Egyptians loved to brag shamelessly so you will find lots of tombs that say things like "I am (name), I am the most beloved of the King because I am so excellent and everybody knows that, particularly the king, which is why he paid for this awesome tomb". Obviously you might wanna take self-assessments like that with a grain of salt, but typically such an assessment is attached to a tomb that probably needed some significant funding.

The reason this becomes hard to answer is we don't get many accounts of such people rising from lower parts of society. That might be because the activities of low ranking scribes aren't well documented, or it might be because it didn't happen and most of the high ranking scribes came from high ranking families.

A good example of a scribe that attained a great deal of prominence was Amenhotep, son of Hapu.