Can I exchange an iPhone 4 with an iPhone 5 without getting a new SIM?

According to this article, the problem isn't just the size.

It turns out, as well as a smaller size, the Nano SIM is reportedly 15% thinner than the Micro SIM cards. So even if users cut the Micro SIM to Nano SIm size with a knife or nano SIM cutter tool, the Sim card would still be too thick to put in the iPhone 5.

short answer: No

The iPhone 5 has the Nano-SIM The iPhone 4 and 4s have the Micro-SIM

iPhone 5 is not compatible with existing micro-SIM cards.

found on the apple site

You have to go to your mobile provider and get a replacement Nano-SIM if you're using the micro-SIM. The other way around you can use an adapter or get a replacement SIM as well.

It could be possible if you haven't used the Nano-SIM you can 'cut out' the Nano as a Mirco-SIM if your provider provides it like that:

There are nano-SIM cutters on the market to cut out a nano-SIM from a normal or micro SIM but I'm not sure these work seeing some bad reviews on the internet... according to Joseph's answer the nano-SIM is also thiner making it impossible to just cut a normal or mircro SIM.

edit: Now I've seen reports that using sandpaper to make the SIM thinner will work. question is are you willing to go trough all the trouble cutting and sandpapering your SIM while you could get/buy a new one at he store that will work 100% ?