When hit by a major health problem, many Americans have seen medical bills eat up not only all their insurance, but also their life savings and other assets.

A compromise plan hammered out in Congress this week to provide catastrophic Medicare coverage for senior citizens would certainly be a big help to elderly Americans. But it also raises a serious question: Can we afford it?The bill is expected to sail through the House and Senate because it is a compromise between two versions that passed overwhelmingly last year.

As it is now, Medicare pays for a maximum of 59 hospital days a year, with a first-day deductible. It also pays 80 percent of doctor charges. Very few drug costs are now covered.

By contrast, the new program would extend Medicare-paid hospital coverage to 364 days a year, and 100 percent of all doctor bills after about the first $1,400. The first 80 percent of doctor fees would still be covered. The 20 percent not now covered would count toward the $1,400 catastrophic deductible. On a gradually-rising scale, all medicines would be covered up to 80 percent, after an initial $580 yearly deductible.

Now get ready for the hard part, the price tag: The catastrophic program is expected to cost $31 billion in the next five years.

Theoretically, the new program would be self-financed by a complex mechanism that includes a $10.20 per month boost in Medicare premiums for the elderly, plus a significant additional yearly tax bite out of any taxable income that a retired person may have. For example, an elderly person with sufficient taxable income could end up paying an extra $1,050 a year in additional Medicare taxes by 1993.

Yet some federal officials are understandably uneasy about the financing mechanism and whether it will produce enough revenue to underwrite a gigantic expansion for Medicare - a program that has been grappling with other funding shortages for years.

The measure raises the possibility of significant expenses that states may have to shoulder. It requires state Medicaid programs - which pay health costs for the poor - to pay all Medicare regular and catastrophic premiums and deductibles for any beneficiaries who fall below the poverty line. There is no way of telling how much that could cost.

In any case, such government programs have a way of growing far beyond any original expectations. It may be perceived as pay-as-you-go at the start, but how long will that last?

As the elderly make up an increasingly bigger proportion of society in coming years, the demand on the program inevitably will increase.

Medicare itself is a case in point. When it was founded in 1965, the federal government spent $3.4 billion on the program, with the outlay in 1990 estimated at $8.8 billion. Yet by 1985, the cost already had topped $70 billion.

The nation certainly must come to grips with the problem of catastrophic illness, particularly among the elderly. But with the federal budget already unable to cope with rivers of red ink, the U.S. simply cannot allow history to repeat itself with another runaway entitlement program.