EDS, which said it had met all the terms of the deadline, claimed Nats had decided three months before the deadline to be "rid of EDS", pull out of the contract and not complete the system.
The dispute could prove to be the IT industry's longest and most expensive court battle.
In suing Nats, EDS is claiming at least £40m in damages and costs. Nats is defending the writ and has issued a counter-claim for £8m.
The contract, the Labour Government's first PFI deal, was for a replacement Flight Data Processing System at the Oceanic air traffic control centre at Prestwick in Scotland. A new system was needed to help control flights across the Atlantic.
Nats claimed that EDS failed to meet the terms of a key contractual milestone in May 2000. But in evidence given by EDS to the Court of Appeal, the supplier claimed that Nats decided to be rid of EDS in February 2000, three months before the May 2000 deadline.
This claim by EDS was part of preliminary legal proceedings in advance of the main hearing, which begins next Monday. Legal costs for Nats and EDS could add up to many millions of pounds, as the case is not due to finish until March.
The Oceanic contract started in June 1997 and was not due to expire until 2011, but in July 2000 Nats cancelled it, insisting that EDS had not only failed to meet a contractual milestone but that it was not satisfied that EDS would meet the final "O" operational date in March 2002.
EDS has won an Appeal Court judgement that allows the supplier to include in its legal arguments the allegation that Nats had decided to be rid of EDS regardless of whether money would be wasted.
Last year Computer Weekly highlighted Nats' antipathy towards PFI contracts, particularly in the run up to the partial sale of the organisation. Nats strenuously denied that its attitude towards PFI had anything to do with the cancellation of the EDS contract.
EDS has never said why it believes Nats cancelled the contract, although it now questions the motivation for doing so.
This week a Nats spokesman said the issues raised in the Court of Appeal ruling could be raised by EDS only as a defence to the Nats counter-claim, not as part of its main evidence. The ruling was unlikely to benefit EDS' case, said the spokesman.