The grave, in County Durham, was the final resting place of Levi George Price, 16, who died in 2001 and Gareth Price, also 16, who died in 2005.
Officers were called to the cemetery on Bridge Street in Metal Bridge near Ferryhill at 11:45 GMT on Saturday.
Police described the thefts as "sickening".
They added the crime "could be linked" to two recent incidents in Darlington in which vehicles appeared to have been deliberately driven into houses.
A stolen lorry was reversed into the bay window of a semi-detached property last week - days after a stolen horsebox smashed into a home on the outskirts of the town.
Police have previously said the two incidents with the vehicles could have been linked to an earlier petrol bomb attack.
Det Supt Adrian Green said: "This has taken things too far. It is depraved and goes against the religious and cultural beliefs of decent people.
"What I'd like to say to the community is this feud has to stop and we need anyone with information about this incident or the wider feud to contact us.
"The incidents to date have been targeted and do not involve the wider community directly."
It is believed the damage took place overnight on Friday.
Gareth Price was found hanging at Lancaster Farms Young Offenders' Institution in January 2005, the day before he was to be sentenced for rape.
He was taken to hospital but died the next day. His death was partly due to failures by agencies, an inquest jury said.
He had pleaded guilty to attacking a girl in the Formby area of Liverpool.
Levi Price also hanged himself, in the garden of the family home in Ferryhill, in 2001.
The skulls of two teenage brothers have been taken from a grave as part of a feud between travelling families, police say.