If bleeding did invalidate fast, our beloved Prophet (Peace and mercy of Allah be upon him) would never undergo cupping while fasting.
Sahih Al-Bukhari
Narrated Ibn `Abbas:
The Prophet (pbuh) was cupped while he was in the state of lhram, and also while he was observing a fast.
Narrated Ibn `Abbas:
The Prophet (pbuh) was cupped while he was fasting.
Narrated Ibn `Abbas:
The Prophet (pbuh) was cupped while he was fasting.
Cupping/Hijama is a process that sucks blood through a small incision made in a person's skin.
If you don't know what cupping is, go through this page from Wikipedia.
However, Hanbali school debates that Cupping invalidates the fast, NOT due to the flow of blood, but because of the temporary weakness it offers. This is albeit a different question you could ask.
One thing you should obviously be cautious is that when nasal or oral bleeding occurs, maximum care should be taken not to let the blood down the throat.
ۚ وَاللَّهُ عَلِيمٌ حَكِيمٌ (And Allah is the All-Knower, the Wise)
اللَّهُمَّ انْفَعْنِي بِمَا عَلَّمْـتَنِي وَ عَلِّمْنِي مَا يَنْفَعُنِي