Unlike my passcard, my R4 doesn't have individual compartments for the connectors, so even a little movement can make it short. Anything I could do about this?
Unless your R4 is extremely poorly manufactured, or you've stuffed the connector in the DS, I wouldn't worry about adjacent tracks shorting when the cartridge is wiggled a little. It's more likely that the spring-loaded contacts are losing contact with the fingers in the cartridge, and that's where the paper shims come in.
There are various ways to do this. My personal favourite is to open up the cartridge and cut out a piece of thick paper or thin card to stick between the board and the inside of the label side of the cartridge. I usually do this with the info sheet from inside Kingston memory card packs - I cut around the edge of the board carefully with a craft knife. I don't use a piece that covers the whole back any more - about half a centimetre beyond the hole is enough - the important thing is to make sure the contacts on the PCB are raised a little further towards the spring contacts in the DS.
Others have reported success with a few layers of tape on the back of the board behind the connector traces, but I'm worried about the adhesive bleeding around the edge of the board or through some of the through-board connections or whatever - ever seen what cheap insulation tape is like a year after it's been applied?
For testing purposes, you could wedge a layer or two of paper or thin card down between the label-side of the cartridge and the slot - again, the idea is to press the edge connector further towards the spring-loaded contacts.
You might also have some success with something like some overhead transparency film secured to the back of the cartridge with something like very thin photo-mounting tape - many people know scrapbookers they could get a small quantity of this kind of tape from, and I often find myself raiding Mrs Arsehat's scrapbooking supplies for various reasons. If I ever have this kind of problem with my TTDS, that's the approach I'm going to have to try because those suckers are glued shut.