Tests were scheduled across India but those at Kamptee, Nagpur, Ahmednagar, Ahmedabad, Goa and Kirkee have now been delayed.
More than 300 candidates are being questioned in connection with the leak, officials said.
In addition, 18 other people have been arrested in Maharashtra and Goa states.
An internal investigation has been ordered, army officials say.
Some reports suggest tests at other centres are likely to be cancelled.
Police said the students were allegedly given the question papers in advance by people connected to coaching institutes.
The exams were for lower-level posts in the Indian army such as soldier-tradesman and soldier-clerk.