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% ?