Added: As Martha and Dusty pointed out, the usage explained in this answer seems to be either incorrect or at best uncommon (thank you for your patience, Martha and Dusty).
The phrase “as many as five” refers to five things with an emphasis on how large the number is. It does not mean “at most five” (= “five or fewer”).
The opposite of “as many as five” is “as few as five.” While it also refers to five things, it has an emphasis on how small the number is, and has the same meaning as “only five.” I think that “as few as five,” “only five” and “no more than five” (as in crypto’s answer) mean almost the same things.