As vernur says, many games are still limited to 2GB for their actual process. This limit can be removed however (there are plenty of guides out there to fix it for skyrim, to take one) and it is likely games will demand more moving forward.
Also, programs other than games have been growing in a rediculous fashion, skype is not afraid to claim 100MB of ram or more at times, browsers are even hungrier. Then there is drivers for graphics, sound, antivirus etc. So unless you want to close all your programs before launching a game, you will need more than 3GB ram.
And you will be limited to about 3GB with a 32bit operating system, the system might be able to allocate 4GB, but that is including RAM on your graphics card and other things. Even mid-range graphics cards are starting to go above 1GB RAM, and I am sure you can see where that will leave you.
EDIT: And as others have said, you can run as good as all 32bit programs in 64bit, except drivers. So you might wish to look out if you have some old macro-keyboard or other peripheral which never got a 64bit driver.