Usually it is believed that CIA is the biggest agency in the world and it follows everything and knows about everyone, but declassified documents about 1971 proves contrary.
In 1971 when Pakistan was busy in fighting Indian backed Mukhti Bahini the USA President Nixon was infuriated because CIA had not enough details about the ground reality and Indian involvement. In one meeting Nixon said to CIA chief:
“We have no doubt that India is involved and that they are probably across the border. But we need something to nail down the exact nature of their activities and we need it in a day or two.”
USA wanted to get involved and solve the crisis before Soviet gains any excess to these countries. At that time USA was more tilted towards Pakistan while India was being supported by Soviet Union (Russia).
In August 1971 Nixon asked whether India will attack East Pakistan (Bangladesh) to which CIA chief said probably not, but at that time India had completed her preparations for an attack on Pakistan.