The blasts happened at about midday (1100 GMT) as the remote town in Borno state was packed with traders from surrounding villages for the weekly market, Chibok elder Ayuba Chibok told AFP.
"Ten died on the spot and another one died on the way to hospital," said health worker Dazzban Buba, who volunteered to treat the injured at hospital.
"A woman and a child died as they were being admitted (to hospital), so now the death toll stands at 13. Thirty others were injured, 21 critically."
Chibok came to prominence in April 2014 when Islamist fighters stormed a boarding school and kidnapped 276 girls, causing global outrage.
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Fifty-seven girls managed to escape in the immediate aftermath but 219 are still being held and have not been seen since they appeared in a Boko Haram video in May that year.
Chibok was briefly overrun by the Islamic State group-allied rebels in November 2014 but recaptured by the military after several days.
"The first bomber set off his explosives at the checkpoint where people coming into the town were being searched," said the town elder.
"A second bomber managed to get into the market and blew himself up. A third bomber was identified and residents pursued him.
"When he realised he was about to be apprehended he detonated his explosives in an area not far from the market."
Buba said he rushed to help his brother who was injured in the first blast in the Bamzir Road area of the town.
But Buba said it was still unclear whether the third bomber deliberately detonated his explosives or whether the device was triggered when troops opened fire as he fled.
Buba said the 30 injured were mostly suffering from burns and fractures, and that nine had been discharged, he added.