I have increased from 3 VPS slices and one core to
VPS with 5 slices, CPU 2Cores, Memory= 5120MB, CPU Storage 125GB, CPU Transfer 5TB.
So, you were effectively running a huge site on 3gb ram and 3 cores? I've been running all of my production sites on a 4gb/4 core VPS and I'm having resource issues at peak times (so I added a CDN).
It really wouldn't matter what software packages or forum software you're running, when you have enough users (including Guests and Spiders) making enough requests (accessing pages, writing data to the database, etc), it's going to chew through resources.
Honestly, I'd expect a site of your size to be utilizing something akin to the servers I shared above... No wonder your host is saying it's UBB.threads, because you don't have the resources available to run a popular community on a tiny VPS account. As a VPS user myself, I'd expect if your site was on one that you'd be utilizing something with at least 6-8gb ram that had been tuned accordingly.
I hope it's an OpenVz VPS vs
kvm; OpenVz, in my experience, is much more friendly with system resources.