What you can cut is really down to which die you use.
Thin dies (sizzlits etc) will only cut paper/card etc
Thick Steel blade dies (Bigz/Sizzix original etc) will "cut almost anything including cardstock, felt, fabric, foam, magnet, leather, craft aluminum, and many other materials." according to the sizzix website.
I have cut felt, fabric, craft foam, leather, chipboard, thin balsa wood (as well as paper & card!) on mine with no problems.
The BS will also cut any make of die (cuttlebug etc)
There is also the new(ish) BS Pro - which cuts up 12"x12" but they are expensive (around £230 last time I looked) as are the dies - just mentioning it in case you were needing this option.
If a machine has not been abused (not sure what would happen if you "force" something throughthat is too think) then there is nothing to go wrong really. You just need to change the cutting mats periodically as they do wear & break eventually (I'm on my 3rd set after about 6 years and I use mine a lot - they are only a few pounds)
The cheapest place online I've found is cuttingedgecrafts I've bought too much loads from them - their P&P is free & they always ship quickly, quite often same day.
Hope that helps, if you have any more questions please let me know
It was just some really thin stuff DH had knocking about in his workshop, so I thought I would steal it and try it, and it worked fine Not sure how thin it was (I used it for cutting out stars to decorate for the local kids group I run, but it was Xmas 2008 or 2009, so I've got none left now to measure...) but I've found this list which says it must be no more than 3/32", which is about 2.4mm if my maths is right.
I think Fred Aldous do balsa wood as I saw it when I was in there the other day (they are an online store too) - but there are probably numerous places you can buy it.