As the player in question who is currently subject to the bleeding effect I figured I'd throw my logic out here.
From a purely RAW perspective there is no exact answer to this question as nothing in Fascinate directly states that suffering damage breaks Fascinate though it is highly suggested.
Some folks have used RAI logic to suggest that some bleed damage is not noticeable or that if the character is bleeding at the start of the fascinate that it does not automatically break it. Now I am pretty sure any sane DM would allow automatic breakage from the fascinate condition if the fascinated creature was stabbed while fascinated. Even if that stab only dealt 1 point of damage it is still a hostile action and threat to the character.
Mechanically speaking, there is no different RAW from taking 1 point of damage from a physical attack and taking 1 point of damage via Bleed damage. Each 1 point of damage is a new instance of damage despite being caused by the same effect.
Examples of Failing Logic:
A. The fascinated creature was immolated prior to being fascinated. By the some of the above logic mentioned, since the immolated character was already suffering continuous damage prior to being fascinated he would not automatically break fascination...... even though he is literally on fire. (A clear and obvious threat)
Now you might argue that bleed damage is more subtle, but RAW, damage is damage whether it be 1 point or 20.
B. The fascinated creature is stabbed in the leg for 1 point of damage and naturally thinks "Ouw that hurt. I should not get hurt or stop whatever is hurting me." (IE: He breaks the fascinate automatically) But as some of the above have suggested it would be. "Ouw that hurt, but look at the pretty colors. I'll ignore this repeating pain."
While fascinate can overrule the characters logic and make them forget about their situation, the RAW rules of the effect specifically call out obvious and immediate threats as breaking it completely. (Not a new save, but complete and utter removal of said condition)
I fully admit that there is room for interpretation on this matter and some of the above points have been excellent in providing alternative viewpoints. In the end this issue has to be decided by the DM.