At least six people have been killed after grenades were thrown at a crowd in Jijiga, the capital of Ethiopia's eastern Somali region.
An unconfirmed number of people were also killed in an explosion in Degah Bur town, officials said.
One eyewitness in Jijiga told the BBC that children and elderly people were crushed to death in a stampede as police opened fire after the blasts.
The government says no one died but blames ONLF rebels, who deny the claim.
Tension has been high in the Somali region since Ogaden National Liberation Front (ONLF) rebels killed 74 people in an attack on a Chinese oilfield in the region last month.
OGADEN NATIONAL LIBERATION FRONT (ONLF)
Want Somali-speaking region to break away from Ethiopia
Founded in 1984
Has been accused of bomb attacks in Somali region and the capital, Addis Ababa
Fought major battles with Ethiopian government in 2006
At least 45 people were injured in the twin grenade attacks in Jijiga. Regional president Abdullahi Hassan, who was speaking at the event, was one of the injured, said government official Bereket Simon.
Five suspects have been detained following the blasts, police said.
Crowds had gathered at the stadium in the regional capital to celebrate a national holiday marking the 1991 overthrow of Ethiopia's former leader Mengistu Haile Mariam.
The eyewitness said it was a chaotic scene, with people screaming and stampeding to get out of the one entrance to the stadium as police were shooting.
Mr Simon also said there had been some deaths in the attack Degah Bur but could not confirm the number.
Sources say the region is now subject to a crackdown by government soldiers who are targeting ONLF fighters.
An aid worker, who asked not to be named, said he had had to apply for permission to enter the affected area, a restriction which he said was slowing down vital development work.
The ONLF says it is not their policy to target civilians.