Sink it will, though, if they lose away to the Suffolk side, Sudbury Town, in tomorrow's first-round tie. While the humiliation for Brighton would not be as extreme as it was in 1973, when as an old Third Division team under Brian Clough's managership they were beaten 4-0 at home by Walton and Hersham, it would still be hard to bear.
Much to the chagrin of their long-suffering supporters, the Sussex seaside club are already something of a laughing stock for football's chattering classes, and defeat tomorrow would make that embarrassing situation far, far worse.
It might be expected that Sudbury's manager is only too happy to hear about his opponents' ongoing misfortunes, in the belief that they give his side a greater chance of victory. Richie Powling is, however, full of sympathy for Jimmy Case, his opposite number, and everyone else who cares about beleaguered Brighton.
"No one likes to see a club in trouble, especially one with such a Cup tradition, but I'm sure they'll turn things round," Powling said.
"Their supporters might not be able to see the light at the end of the tunnel, but when a club hits rock bottom something normally happens to pick them up again, and I'm sure that will be the case with them."
However, Powling of course hopes that Brighton's fortunes do not take a turn for the better after until this weekend. He is realistic about his side's chances of success, though. "If we're honest, eight times out of 10 they're going to beat us," he said, "but the Cup is about dreams and banana skins and we just hope that this might be one of them."
"For the neutral fan, this is one of the ties of the round," Powling added. "They're not having a good season, with all their troubles, and they could be there for the taking. We've obviously got a chance."
Powling knows something of the highs and lows of football. As a tough little wing-half and an England youth international, he signed professional forms for Arsenal in 1973. Over the course of five seasons at Highbury, however, he only managed to appear in 55 League games, before a serious knee injury ended his playing career.
Powling moved on to Barnet, where he started learning the managerial trade under Barry Fry. After spells in charge of Tiptree and Harwich, two Jewson Eastern Counties League clubs, he has been manager of Sudbury since 1992. Now 40, he is employed full-time by the Dr Martens League Premier Division club.
Powling's squad has several players will Football League experience. Tony English (who is also assistant manager), Nicky Smith, Steve Ball, Clive Stafford and Michael Cheetham have all played for Colchester. The tall centre-forward Christian McClean has been with Northampton, Swansea and Bristol Rovers, while his striking sidekick Ian Brown has turned out for both Birmingham and Bristol City. However, English and Brown are struggling with injuries - as is the first-choice goalkeeper Steve Mokler, so 19- year-old Dave Walton, released by Watford in the summer, may earn a chance to make a name for himself.
Although this is their debut appearance in the first round of the FA Cup, Sudbury have played at Wembley more recently than Brighton. They went there for the FA Vase final (which they lost to Tamworth in a replay at Peterborough) in 1989 - six years after Brighton's FA Cup final. In 1990 Sudbury were promoted from the Eastern Counties League to the Southern (now Dr Martens) League, in which Powling took them up to the Premier Division in 1994.
It will be hard to rise any higher, though. "We're the paupers of this league," Powling explained. "Sudbury's only a small little market town, so we struggle. We can't afford to get involved in the transfer market."
Although their Priory Stadium is picturesque, it is inadequate for anything loftier than their present level. Sudbury are awaiting planning permission for a new stadium, which they hope to start building within the next two years - so a continuation of this season's Cup run will provide much-needed revenue as well as much- appreciated prestige.