Test scores and marked papers were originally supposed to be with schools by Tuesday last week. The National Assessment Agency then changed the deadline to this Tuesday after problems at ETS, the firm marking the papers.
Ken Boston, the test regulator, told MPs on Monday that all key stage 2 marking was complete, but schools have emailed the National Association of Head Teachers saying they have missing papers in whole subjects - typically English.
The emails, seen by The TES, are among more than 100 to have been received by the union since Monday. One school, in Bath, said its English papers had turned up at a primary in Milton Keynes last Friday in a bag of that school's science papers.
Others, despite having most of their papers back, have some children missing on the online test results database.
Other complaints include pupils being marked absent when they took a test, different markers working on the same paper and many concerns about the quality of marking.
Meanwhile, a key stage 3 reading marker told The TES this week that papers from three schools were still sitting on her lounge floor.
'I HAVE NO FAITH IN THE SYSTEM'
Initially, Moss Side Primary did not seem to be affected by the delays in this year's test results. The school, in Leyland, Lancashire, had the papers returned for its 35 Year 6 pupils on Friday July 4 - before the initial deadline of July 8, which was later extended to July 15.
But Janis Burdin, its head, is not happy. She is sending all 35 writing papers back to be re-marked.
She said: "They have been so badly marked. It was immediately obvious something had gone wrong. As a teacher, you can list the approximate order the children will come in when ranked in terms of scores. This year, the writing scores bore no relation at all to how we would rank the children. Reading scores bore more of a reflection on how we would rank the children, and science and maths were OK.
"They promised us the standard of marking would be no worse, but we feel it is a lot worse. I have no faith in the system."
This year, children were asked to write a fictitious biography of Pip Davenport, a Victorian inventor, and a short page on "Memories of the School Year".
Mrs Burdin said: "It is very easy for heads to say this should have got better marks ... Some of our papers have been marked too highly, as well as some too harshly. I'm declaring this needs to be marked down. That's how strongly I feel."