'Assuming it's paired with the right CPU...' Of course with sufficient cpu, the GeForce 9600 GT will play any 1080 video smoothly, I suspect.
I believe the 9600 GT was considered a decent video-playing card for its time. It provides 'PureVideo 2', like most of the later GeForce 8000 cards -- apparently this means 'complete' acceleration of H264 and 'partial' acceleration of other compressed video (eg MPEG2, VC1). I should think the vast majority of 1080 content available online is in H264 (eg your typical .mkv file ripped from Blu-ray).
The 9600 GT was available in several designs, and the original reference design (which I own) had a single-slot cooler that yielded relatively high heat and fan speeds. This might disqualify it for many HTPC builds. But I expect you're already familiar with the noise and heat levels of your old card...