The two councils, Oxfordshire CC and Cambridgeshire CC, have had their budgets rejected with no concessions.
But the only other spending rebel, Merseyside Fire and Civil Defence Authority, had its budget accepted.
Three other authorities - Essex CC and Manchester and Tyne and Wear FCDAs - which breached their cap for technical reasons had their budgets accepted.
Cambridgeshire faces spending cuts of £5.7 million to stay within cap. This would reduce its average band D tax by £39.
The county had wanted to spend the extra funds on schools and set the budget only after winning public support through extensive consultation.
'If the government insists on enforcing this spending limit, it will mean fewer teachers in our schools and larger class sizes - again,' said Liberal Democrat leader Peter Lee.
Oxfordshire faces cuts of £7.4m, which will lead to a reduction in its band D council tax of around £35.
'The government can't justify heaping cut upon cut on the people of this county by saying Oxfordshire CC wastes money or is inefficient. Its own figures published in March show we aren't,' said Labour group leader Bob Langridge.
The councils face billing costs of nearly half a million pounds. They have 28 days to appeal Mr Gummer's decision.
Merseyside FCDA will be allowed to spend £52.8m, £2.1m above its provisional capping limit. This follows a threat from the chief fire officer that the authority would not be able to meet its statutory duties.
Mr Gummer had to formally cap Essex CC because of an error in last year's budget round. The county discovered a mistake in its 1995-96 education standard spending assessment and it was not discovered early enough to increase its budget. But the county has been allowed to set a budget £3m above its £971m cap to correct the error.
The two other fire and civil defence authorities were allowed to exceed their capping limits since they had been unable to take advantage of a last-minute changes to 1995- 96 rules which came after they had already set their budgets.
Four out of 10 authorities breaching their provisional capping limits were allowed to increase their budgets last year. Three were allowed a rise after appealing.