The court in Banda Aceh, on western Sumatra island, Thursday handed Chief Sergeant Joko Rianto a two-month jail term and Chief Private Rawali a three-month sentence.
Rianto was given a five million rupiah ($460) fine while Rawali, who like many Indonesians goes by one name, was ordered to pay 2.5 million rupiah.
"Rawali and Joko Rianto have been found legitimately and convincingly guilty of illegally possessing dead protected animals," judge Lieutenant Colonel Budi Purnomo said.
Rianto, who was caught with one of the tigers and a bear in his house, argued he had purchased the critically endangered tiger to use its teeth to cure his sick wife.
Tiger parts are frequently used in traditional medicine in Asia despite the lack of peer-reviewed scientific evidence showing that they have any medicinal benefits.
Rawali claimed a friend had given him the tiger to repay a debt.
Ratno Sugito, a local animal activist, welcomed the sentences: "Even though the sentence was weak, at least the military court showed its willingness to enforce the law."
It's believed to be the first time Indonesian soldiers were convicted of illegally possessing dead protected animals.
Sumatran tigers are the most critically endangered tiger subspecies. There are only an estimated 400 to 500 still alive in the wild on the island from which the animal takes its name.
Its numbers are rapidly dwindling due to destruction of its rainforest habitat and poachers targeting the animals to sell their parts, mainly for use in Chinese medicine.
The court did not disclose the species of the bear although it said the animal was protected by law.
Voice of Russia, AFP, abcnews