What is the relationship between Turing Machines with a finite tape and Finite State Automata?

I seem to recall from an undergraduate class that for a Turing Machine with a finite tape there will always exist a corresponding Finite State Automata, but I've been unable to find this confirmed anywhere on the internet. Is this actually the case or am I misremembering?

• How many possible states will a Turing machine with a finite tape have? – Dave Clarke Mar 31 '15 at 16:21
• It will be finitely many but, as the below answer shows, that's not necessarily sufficient for drawing an equivalence. – Jesse Berlin Mar 31 '15 at 16:27

But if you mean a bounded tape, where the tape has $k$ cells for some fixed $k$, then yes - you indeed get a model which is equivalent to DFAs.