A level 15 monk has 4 attacks when using bare-hands.
Is it possible to do 4 stun attacks in one round, or is only 1 stun attack possible and 3 normal attacks?
Without taking other feats only one Stunning Fist attempt can be used per round.
According to the 3.5 Player's Handbook a creature with the feat Stunning Fist
Emphasis mine. Being a monk changes this to once per day per monk level but doesn't change the frequency per round of Stunning Fist attempts.
The 3.0 Player's Handbook reads, "The monk can use this ability [stunning attack] once per round, but no more than once per level per day" (PH 39). Emphasis mine.
The 3.5 feat Rapid Stunning (CW 104-5) grants the character the ability to make 1 additional Stunning Fist attempt per round; the feat can be taken multiple times (e.g. a character who takes the Rapid Stunning feat once can make 2 Stunning Fist attempts per round, a character who takes the Rapid Stunning feat twice can make 3 Stunning Fist attempts per round).