Many people who have failed to pay taxes on funds stashed in overseas bank accounts will likely toss and turn during coming nights, worried that the tax man will soon come knocking at the door. Will they be among the nearly 4,500 account holders whose names Swiss bank UBS has agreed to give to the IRS? And even if their names aren’t on the list, will the IRS learn about them from others seeking amnesty? Should they apply for amnesty themselves, paying large tax bills but at least staying out of jail? Or lie low for fear the IRS will find other problems if they draw attention to their returns?

First, a little history. A few years ago, a disgruntled or guilt-stricken (depending on who’s telling the story) American who had worked at UBS went to the IRS with a story about secret bank accounts. The bank, he claimed, had sold to investors techniques that let them hide foreign accounts from government view. Many rich Americans bought into the plan and failed to report account income on their tax returns. That was no simple oversight; anyone reporting interest or dividends on Schedule B has to check a box saying whether she had a foreign bank account.

The IRS initiated a three-pronged attack. It began investigating taxpayers whose names had been revealed, sued UBS, and demanded that the bank identify all American account owners. At the same time, it set up an amnesty program that required taxpayers with unreported foreign income to pay back taxes, penalties, and interest but allowed them to avoid stiffer penalties and potential jail time.

Those strategies yielded a bountiful harvest. The IRS has successfully prosecuted some evaders, in one case collecting $52 million in back taxes. As part of a $780-million settlement with UBS last February, the bank promised to hand over the names of 250 account owners. Under a separate agreement announced yesterday, the IRS will get another 4,450 names.

Those moves really juiced the amnesty program. Facing a September 23 deadline, tax evaders have lined up to confess their sins. Last month, the IRS reported, more than 400 showed up in one week, “four times as many as in all of last year.” You can’t request amnesty once the IRS has started investigating you, and even with amnesty, you’ll still owe Uncle Sam a lot. But amnesty can make a big difference. The IRS offers this example: for six years, a taxpayer fails to report $50,000 of annual earnings on a $1-million account. Request amnesty and you’ll owe the IRS $386,000 in back taxes and penalties, plus interest.  Wait for the IRS to come knocking and the tab could jump to $2.3 million plus interest and probably jail time. Not a hard choice if you think the IRS is on your tail.

The New York Times reports that UBS will let the 4,450 account holders know before September 23 that the IRS will get their names. That’s time enough to claim amnesty. But what if you don’t hear from UBS? What are the odds you’re not on the list?

And then there’s the section in the IRS’s suggested “voluntary disclosure form” that asks about communications with banks and financial advisors concerning overseas accounts. Someone else’s disclosure might finger your advisor and the IRS finds your name in his records. If you dealt with one of the major brokers, amnesty sounds like a pretty good deal.

But turning yourself in has its own downside. You’ve bumped the probability that the IRS will look at your returns to 100 percent and who knows what else they might find. Amnesty may protect you against huge penalties on those UBS accounts but you’re still on the hook for anything else the auditors find. So maybe you want to report other accounts you have with other banks. Who knows where that leads? Maybe you’re better off lying low and taking your chances.

The UBS case is one more skirmish in the continuing battle between tax evaders and the IRS, with this round going to the tax man. But it’s just one round. Remember the flap a few years ago over people using credit cards to access money stashed in foreign banks? The IRS got tens of thousands of account holders’ names but couldn’t cope with the numbers and the big bust seemed to fizzle. Nonetheless, shining more light on secret accounts can only make life tougher for tax cheats.

Meanwhile, a lot of people will battle insomnia over the next five weeks as they decide what to do. Of course, they’d sleep a lot better if they’d just paid their taxes in the first place.