It's Worse Than That
Actually though, the situation is even worse for them than just "not making money". If you consider inflation, which you of course should, everyone receiving zero nominal return are in fact experiencing a decline in the value of the money they saved, meaning that they are in effect giving away part of their savings to the institution they lended it to.
That is why, as I've written before, the only thing safe about certain "safe havens". like government securities issued by the American and British government is that anyone who invests in them is going to lose.