All of them were out of danger after receiving medical treatment, the official said.
The incident took place at a government middle school in Panchmandia in Vaishali, about 50 km from here, a district official said.
"Soon after drinking water during the break, the children complained of uneasiness and started vomiting. At least 19 of them fell ill," the official said.
"The children were taken to a government hospital in Lalganj and were discharged after treatment," he said.
Some children told police that they could smell the presence of some chemicals in the water which they fetched from a hand pump.
Last year, several such incidents were reported across Bihar, forcing the state government to order an inquiry.
Last month, students of a government primary school fell ill in the state's Bhagalpur district after eating a mid-day meal in which a dead lizard was found.
According to education department officials, there have been 14 such incidents in the past seven months, including a case from Bhagalpur.