"The humanitarian situation is getting worse despite the scope of the operation increasing," said Peter Maurer, president of the International Committee of the Red Cross (ICRC). "We can't cope with the worsening of the situation."
The ominous announcement comes as beleaguered Syrian President Bashar al Assad reaffirmed to the world on Russian television he would not seek safety in exile.
"I am not a puppet made by the West to go to the West or to another country. I am Syrian... I must live and die in Syria," Assad said. "I am a Syrian. My home is in Syria."
Maurer also said there were a number of "blank spots" — areas in Syria where the Red Cross could not reach victims. Aleppo, a city that has seen hard, sustained fighting, is one example.
"There is an unknown number of people in Syria who do not get the aid they need," Maurer said.
The Syrian Observatory for Human Rights claims 37,000 people have died since fighting began in March 2011.
The UN's refugee agency, the United Nations High Commissioner for Refugees, said, "The Syrian Arab Republic hosts one of the largest urban refugee and asylum-seeker populations in the world." They have registered 305,069 refugees, though there are likely more.