The risk of losing money in a savings account is not zero. By losing I mean, losing its purchasing power. Banks have repeatedly gone bust throughout all of history, in all countries. But there is such a thing as deposit insurance. In the U.S., your savings (up to 100K per account) are 'insured' by the FDIC, but the FDIC has a mere pittance of money compared to the total deposits in all the banks, or even in one of the big banks. It cannot, using the money it has, save all depositors if a big bank is allowed to go bust.

So, given your bank goes bust in say, one of those recessions the U.S. experiences in greater and greater frequencies, either you lose nearly everything in your 'savings' account as the FDIC doesn't have enough money to cover all of its deposits the bank loaned out for its own profit - fractional reserve banking serving YOU since 1913. OR the FDIC pleads to the Federal reserve to 'give' it money, print it that is, causing massive inflation. Though in that latter case, you get the money first, so get to spend at current prices before the influx of new currency causes prices to inflate.

Though in current times, the solution is that these banks are too big to fail. So whenever they gamble your money to make a profit, yet lose, they get some of those nice big bailouts from the Fed. In that case the banks get to the spend the money first, and everyone else holding USD gets an inflationary hit - again you lose your purchasing power of your savings.