I've got a 4K monitor attached via Dual Link DVI. It does 4K, but only at 30Hz.
It does 60 Hz up to 2560x1440; anything above that is 30Hz.
Is that a limitation of DVI?
A single dual-link DVI connection can deliver up to 2560 × 1600 at 60 Hz. Achieving 4K at 60Hz requires two dual-link DVI cables, or DisplayPort 1.2 or HDMI 2.0.
So, yes, as you put it, that's a limitation of DVI.
No, this is beyond DVI spec and I haven't been able to successfully force it.
You can, however, achieve 2K (2560x1440) @60Hz over a single dual-link DVI cable, and you can achieve 4K (3840x2160) @30Hz over a single dual-link DVI cable.
I have successfully tested this on a Geforce GT 730 connected to an Asus MG28UQ 4K monitor with a DVI-to-HDMI adapter cable.
My colleague Viktor Klimo has achieved 4K @30Hz with dual-link DVI out of an Nvidia GT 730. It's a great and affordable solution if you can live with 30Hz (30Hz for some reason looks better on some monitors than others, looks terrible on 4K televisions when I tried 30Hz via HDMI but looks okay on my Eizo EV3237).
I did some experimenting with HDMI 4K@60Hz but it was a lot of trouble even with special adapters. For 4K@60Hz only DisplayPort works reliably. In the future with HDMI 2.2 the situation will improve.
I have a Dell 3007wfp-hc monitor using DVI-D dual link 2560X1600 native and am on a ASRock z68 Extreme4 Gen3 / 32GB RAM/ i7 2600k oc to 4.4GHz. I have 2 X ASUS Strix GTX 980 SLI enabled with DSR selected. most of my games are running 3840X2400 at 60fps minimum and the few that dip below 60 fps don't go below 40.
My gaming platform is not supposed to be able to run 3840X2400. Obviously the reviewers are telling us something about a compelling need to upgrade to achieve a quality 4k gaming experience. What it is may not be the whole story since my experience on a 5 year old platform clearly are not exceptional beyond the fact that I am using factory OC video cards and have a modest OC on my Sandy Bridge. OS is windows 10.