A 5870 or 5850 should be able to run Crysis at or above 50 FPS (which should present no visual stuttering) at Very High with no anti-aliasing applied. Using these performance charts
we can see that the HD 5870 and HD 5850 run it at a Very High setting with two times Anti-Aliasing using the 1280x1024 resolution at 40.6 and 48.5 FPS respectively. If you disable Anti-Aliasing you should see at least a five FPS increase on both fronts.
Crysis Warhead is a little more optimized and will provide possibly another four or five FPS increase over the original Crysis at comparable settings.
Some may argue for 1440x900 resolution 1680x1050 is a better judge of performance, but a 1280x1024 resolution has 2304 pixels, and 1440x900 has 2340 pixels; that is only 36 pixels away as opposed to 1680x1050's 2730 pixels (426 pixels away).
A note on this performance however: The benchmarks were done using a Core i7 machine with DDR3 memory. Your Core 2 Duo will not provide comparable CPU results and your memory will not provide comparable memory results creating a disparity that may dip your total performance a noticible amount of frames.