Some voters queued for more than three hours
Liberal Democrat leader Nick Clegg has said he shares the "bitter dismay" of voters who had to wait in long queues at some polling stations in Sheffield.
Many people at the Ranmoor station in Mr Clegg's Sheffield Hallam constituency queued for more than three hours, with some unable to vote.
"That should never, ever happen again in our democracy," Mr Clegg said, as he was returned as Sheffield Hallam MP.
The Electoral Commission is to carry out a thorough review of what happened.
Police were called to move people who staged sit-in protests at three polling stations in the city on Thursday.
Officers were called to a polling station at Bents Green Methodist Church - near Mr Clegg's house - where several people were protesting after being unable to vote.
John Mothersole said the council was "caught out" by high turnouts
Police were also called to Ranmoor, where about 200 people were turned away from voting, and a polling station in Woodseats to move voters who were refusing to leave.
The Electoral Commission said it would be undertaking a "thorough review" of what had happened in constituencies where people were unable to vote.
John Mothersole, Sheffield City Council chief executive and returning officer, said: "We got this wrong and I would like to apologise.
"We were faced with a difficult situation with the numbers of people, and a large amount of students turning up to vote without polling cards.
"This made the administration process of ensuring the correct person was given a ballot paper much longer.
"The only remedy, which we could not take, was to extend the voting times."
Dr Matthew Winter said he was turned away from the city's Eccleshall Parish Church polling station at 2200 BST.
"I am frustrated I have not been able to vote.
"I think it's wrong there were not enough people there, there were only three people there who were processing votes and giving out ballot papers."
Former Hillsborough MP Helen Jackson said the situation was disgraceful and could result in a legal challenge if results were close.
BBC reporter Tom Ingall described the scenes at Ranmoor as "extraordinary".
Boundary changes have seen two newly-created seats of Penistone and Stocksbridge, and Wentworth and Dearne.
The Sheffield Attercliffe seat has now been renamed Sheffield South East. Sheffield Brightside and Sheffield Hillsborough have become one seat.
The changes were part of a reorganisation by the Boundary Commission for England.