I've just given my old 2560x1600 monitor (Dell UltraSharp 3011 or 3011t) to my parents. Their PC has an Intel mATX early Haswell motherboard, I'd guess about 2013 - 2014 ish? It's got DVI and HDMI ports, I haven't checked if the DVI is DVI-D/I.
When I connect the monitor to the board, it can't go higher than 1920x1200, reporting that the bandwidth needed for the video connection is too high, which is fair.
Can a better HDMI cable or a dual link DVI cable handle 2560x1600, or do I need to plug in a discreet video card?
(If it needs a new video card, that would probably be DP which would also solve it)
I'm sorry, not being there I can't look up the exact motherboard model. But I'm pretty sure its haswell ~ 2014 ish, Intel own-brand on board or on-chip GPU (hence not AMD/nVidia), and can add the correct model later.
Mostly, I can't find the bandwidth/resolution limits for that era HDMI/DVI, and that'll probably be what determines if the hw can cope with a different cable or needs a card update.