It looks like it is a CIBC call centre and/or possibly a CIBC collections number, which is why they won't tell you what department they are. What an absolutely stupid hassle they cause by acting so suspicious like this. As of July 2010, they do answer the number. I was called by 416-784-5900 and a women on the phone told me they are a call centre located at 750 Lawrence Ave M5L 1A2. I have no way to confirm that information though.
But they called me out of the blue and it started like this "Hi, John Doe? Great. In order to cofirm you are who you say you are, we need you to answer the following security questions". I then responded, "wait, you called me. How do I know you are CIBC? How about I call you back?", response was "Ok, you can call us back at this number 1-800-597-5525". To which I responded, "That doesn't sound right. How about I call CIBC main branch, and have them transfer me to your department? What department do you work in?", the response was "we can't give you that information".
It seems suspicious except that 1-800-597-5525 is the CIBC student loan number.