We run 1Gb over Cat4 at work LOL. Cat7 is what? 10Gb?
Yeah it's a mental speed for cat 7, but it's dependent upon the kit being used, length of cable, switches etc. I've worked in a 10Gbps environment, but the only thing that can perform at that speed is storage or very high end switches (Nexus) for instance.
TBH if you're going for 10Gbps speeds you'd normally use fibre cables anyway, and have a fabric network.
So from what you've said it sounds like a cable issue, or NIC issue eg the NIC cannot perform at those speeds. Depending on the NIC i'd be surprised if it couldn't handle a min of 1Gbps.
As ketmp said, it's a process of elimination, but it's likely the cable, you may have an earth or something. Without a cable tester it's hard to judge. You can make your own cables quite easily, just get some decent cat5e cable, and some RJ45 connections and cut and fit your own cables.
Most cables are STP (Shielded twisted pair), try and buy roche (Think that's what they're called) cables, as they're pretty decent.