DiscretePlot[Mod[n !,2^n],{n,20,35}]
Plot[Mod[m !, 2^m], {m,20,35}]
What do you mean with 'fail' ? x! grows as x^x, so much slower than 2^x, so the 'mod' will chop up the function many many times and it will look like a lot of vertical lines... The values DiscretePlot shown (the integer ones) might not appear exactly in the Plot version, as it samples PlotPoints number of samples which might not be exactly integers...
Edit: Unfortunately we can not see the notebook as we don't have permission to do so. Therefore I'm speculating. Can you provide a screenshot?
