The architectural firm, currently working on New YorkвЂ™s 11 September museum, was kicked off the job to design the Margate Art Gallery after three years of design work.
It won the contract in conjunction with Spence Associates to design the JMW Turner gallery in 2001 with a vision to build a timber clad concrete structure in the sea and link it to Margate by a steel bridge.
It had promised the council it could work within a ВЈ7m budget for building work or ВЈ12.6m exclusive of agreed client charges.
But as designs progressed after SnohettaвЂ™s formal appointment in December 2003, the estimated construction costs soared.
In February 2004 Snohetta decided to change the building to a steel structure with a steel shell after concerns timber would not withstand the marine environment.
Following a tender that put construction work at ВЈ28.4m and ВЈ12m for work to be carried out by a subcontractor, Kent County Council scrapped the project in February 2006.
It appointed David Chipperfield Architects to redesign the building, which received planning permission earlier this year and will cost ВЈ17m.
In a writ filed at the Technology and Construction Court, the council is seeking ВЈ5,943,815 plus interest from Snohetta over the failed design.
It claims the firm failed to appreciate the actual cost of changing the design to steel, did not properly design or appreciate the cost of the galleryвЂ™s foundations and failed to realise how much achieving a smooth steel surface вЂ“ as promised вЂ“ would cost.
The council also alleges Snohetta did not consider how the 2,000 tonne structure would be installed вЂ“ particularly since there were no mobile cranes that could lift more than 1,200 tonnes in the UK at the time.
A contractor to build the revised art gallery is yet to be announced.