They said they had seen pupils sobbing. They recorded cases of children who felt sick and of others who refused to go to school. Parents were adding to pressures by promising their seven-year-olds gifts if they did well in their tests also scheduled for this month.
Their comments came as ChildLine, the counselling charity, warned that children's anxieties should be heeded.
Hereward Harrison, its policy, research and development director, said:
"Parents should make sure children are leading a balanced life and not studying all the time."
About 600,000 primary pupils took tests in maths, science and English last week.
John Illingworth, head at Bentinck primary in Nottingham, said: "One child came into school in tears worried about these tests.
"Last year, I had a family whose younger child had been killed in a road accident. They were offered a cheap holiday during national tests week. The mother came in to tell me and I remember saying "yes, go and enjoy yourselves".
"But their daughter was outside crying because she didn't want to let the school down. That child was a level 4, possibly a level 5."
Graham Eskell, head of Montgomery junior in Colchester, said: "We had one lad who was sick every night. To have this pressure put on 11-year-olds is shocking."
A Blackburn teacher spoke of two Year 6 children who refused to go to school and three children who cried during the tests and said: "There is something sadly wrong when you have to put children through this."
One Midlands head said: "I had one girl last year so scared stiff that she absolutely lost it, there is nothing you can do. In the end I took her out of the test because she was so distraught."
John Bangs, head of education for the National Union of Teachers, said:
"These are alarming tales, which must be common. It is a shame that in Year 6, when there is the opportunity for children to get involved in challenging projects, they are all cramming for tests."
A Qualifications and Curriculum Authority spokesman said that the authority had no evidence that pupils find the tests stressful, which reflected the effort put into ensuring tests were "appropriately challenging and enjoyable".