At least 15 people were killed in a powerful blast that ripped through an Independence Day parade in the northeastern Indian state of Assam.
And in Indian-ruled Kashmir another 15 people, including some children, were hurt when suspected rebels fired a grenade during festivities at a high school, officials said.
"We must stand on the twin pillars of mutual trust and confidence. But of course, cross-border terrorism and violence make our task more difficult," Prime Minister Manmohan Singh said of the country's relations with neighbouring Pakistan.
Dr Singh's speech came as festivities were marred by the powerful blast in Assam, the victims of which included seven children, police said.
Dr Singh (C) has vowed to carry
on peace efforts with Pakistan
Guerrillas fighting Indian rule in the remote oil- and tea-rich state had called for a boycott of the independence celebrations.
A senior police official said the conditions of some of the wounded were very critical with multiple injuries in their chest and faces.
About 150 people were watching a Bengali language film when the blast took place.
"There was total panic after the blast, with many people injured in a stampede that followed the explosion," he said.
Hours earlier, rebels blew up a crude oil pipeline in eastern Assam's Dibrugarh district.
A senior police official in Guwahati blamed the outlawed United Liberation Front of Asom for the two blasts.
"We must stand on the twin pillars of mutual trust and confidence. But of course, cross-border terrorism and violence make our task more difficult"
Dr Manmohan Singh,
Indian prime minister
In Indian Kashmir, meanwhile, 15 people, including some children, were hurt when rebels fired a grenade during festivities at a high school in northern Baramulla district, officials said.
Rebel violence in the scenic Himalayan region has escalated in recent weeks despite ongoing talks between the nuclear-armed rivals which both lay claim to Kashmir.
New Delhi has long accused Pakistan of fomenting the 15-year insurgency in Kashmir, a claim rejected by Islamabad.
More than 50,000 troops were put on alert across the seven northeastern states, with 10 separatist groups calling for a boycott of the Independence Day celebrations in the region.