I ran into this problem today and thank goodness I found the thread. I was playing with my new iPhone 5S and swapping SIM between the N4 and iPhone 5S. As I was putting the nano sim adapter with the SIM back in, I forgot to hold the SIM with my hands and it fell out as I slid the tray back in. Whoops. For N4 users using this Silverhill adapter, be wary--the nano SIM adapter falls out one way from the main tray, and the nano sim itself falls out the other way. You need to make sure that BOTH are in place before sliding the tray in.
3rdeyer's instructions worked pretty well. I'd like to add some details.
First of all, I inserted the plastic strip UNDERNEATH (when the screen is facing up) the sim tray and just jammed it in as far as I could. I used only 1 strip and I have a photo here to show what the dimensions are. You don't need 3cm of length because it won't go in that far, but it's a matter of having something for your fingers to grab onto. So what I'm showing is probably more like 2cm. I made mine a little less wide than the SIM tray and it worked wonderfully.
One thing to note that as you jam the plastic strip in, it seems the SIM tray slides inward a little too. But do not fret! Once it won't go in further, start pulling out the SIM tray. Pull it out in conjunction with the plastic strip. You won't be able to pull it all the way out, but that's fine. It'll be mostly out where you see the nano SIM tray, and you can slide the nano SIM in. Once you slide it in, you can pull the entire TRAY out now. Finally.