You can use the wafer thin dies (Nestability style) to emboss into cardstock rather than cutting through for a nice finishing "frame" on a design. Sample attached and
here's a link for a "how to" in the tutorials section (Lori is using a Wizard in the tutorial but the same principle applies for other machines).
As Scrapjanny says, steel rule dies and a manual cutter are great for thick materials - I'm still working my way through the bag of mountboard (mat board on your side of the Pond, I think) I got for free by asking a framing shop if they had any spare offcuts.