Did you uninstall the previous display driver before installing your graphics card driver? Did you restart after installing the new card's driver?
If the answer to both questions is yes, download (or copy from somewhere) the latest ATI driver and install that. Alternatively, install your monitor's driver. Either of these two approaches will probably solve the problem.
Monitor has hardly got anything to do with the resolutions shown. It is the fault of the graphics card. Uninstall the graphics driver already installed and install the latest drivers available. The resolution should show up.
If it doesn't, buy a DVI cable. Doesn't cost much (~200). Might help.
These days, even cheaper boards' IGPs support those resolutions, no reason why your graphics card won't
Glad to know your problem is solved. But are you sure it was the cable?
Last week, I assembled a new machine for a client using a Samsung B2030 monitor and an HD 5670 graphics card. The gfx card has only a DVI output while the monitor has both VGA and DVI inputs, but came with only a VGA cable.
So I used the VGA cable with a DVI-VGA adaptor at the gfx card output. No problem at all with the display at the native resolution of 1600x900.