I've checked the forums and couldn't find a thread about this bug, forgive me if I missed it.
I just started playing and noticed that the bank at tutorial island doesn't work. I couldn't 'bank banker' or 'talk-to banker'. When I finished the tutorial I tried using the two banks in Varrock and the one in Draynor Village. Both didn't work either. I can 'examine' the banker but that's it. It's very anoying and I was wondering if anyone else is having this problem or that the banks I checked just aren't working yet.