Indian soldiers are using boats and helicopters to reach several hundred remote villages.
The flooding occurred as water flowing from Nepal caused the Kosi river to breach its banks and change course.
The BBC's Sanjoy Majumder in Bihar says more rain is expected in the next two days so authorities are moving as swiftly as they can to evacuate villages before the waters rise again.
The continuing bad weather is hampering efforts to get aid to about 2.5 million people who have been displaced.
'National calamity'
Our correspondent says many of those stranded in remote villages are sitting on the roofs of their submerged homes.
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Tens of thousands of people have crowded into temporary relief camps.
The boat incident took place on Friday in the worst-affected district, Madhepura, 150km (95 miles) north-east of Bihar's capital, Patna.
Police said 40 people were saved.
ON Bhaskar, superintendent of police, told the Associated Press news agency: "The boat was overcrowded because people panicked to be rescued and clambered on board."
Angry villagers in Madhepura said they had no idea where to take shelter and complained they had received no food or aid.
"We have lost everything," said Bimlesh Yadav, escaping with his family to a nearby town.