It is possible; but probably not at a price you want to think about.
http://www.foresightimaging.com/PRODUCTS/hdp/accustream170.htm has a capture card that'll manage it for only $2,995. Also, 1080i over DVI has a bitrate of 1.78gigabits per second(if I remember correctly). 8 bits in a byte gives us c. .222gigabytes/second. or 222Megabytes/second. You'll probably need a a 66mhz/64bit PCI bus just to transfer that, and I shudder to think of the RAID array you would need to store it.
By contrast, the SGI Scalable Graphic Capture solution (compatible with the modest
http://www.sgi.com/products/visualization/onyx350/ Onyx350) makes the above option look positively sane and sensible.
Slashdot has a piece on the question
http://ask.slashdot.org/article.pl?sid=04/11/18/0233207&tid=129 . So far as I know, HD record is Just Not Happening with consumer gear right now. Pity, really.