That is what most people think and this is not what happens when people save money; see Fractional Reserve Banking.
http://en.wikipedia.org/wiki/Fractional_reserve_banking
To me, this doesn't contradict the lending theory, it means your dollar in a bank is loaned several times over (what could possibly go wrong?).
Not that this affects anyone's answer to the original question.
FRB just makes bankers more rich. From what I understand of fractional reserve banking, the bank does not loan your money to other borrowers as many people view. The bank borrows from the central bank and pays it as a loan - in the UK, there was not long ago the LIBOR scandal and banks fixed their rates between other banks. The money passed on to the borrower has thus an added interest fee plus the effect of inflation which is viewed as a hidden tax; i.e. your money is worth less over time.
The problem with FRB is that if there was a bank run, the banks would have a very short supply of notes as these are no longer backed by gold or silver. It is just a very good scheme to make paper (the IOU known as greenbacks during the american civil war) as valuable as gold. Now in the advent of electronic entries, it can create another problem that the figures that is represented on your bank statement is not even backed by paper.
Going back on the OP, I would spend the interest because interest rates are too low to save and I have no confidence in the banking system to keep most of my money stored in banks.