No, this can not be done with the Sticky Notes utility that comes with Windows 7. That utility is hard coded to look for only one sticky notes file in a certain APPDATA location. So even if you could start multiple copies of it, it will open the same file.
The only way you can have different sets of sticky notes with Windows utility is to switch to different userid in Windows temporarily. This way, each login will have its own sticky notes. But that solution is inconvenient.
Otherwise, you need to look for third party Sticky Notes applications that support having multiple sets of sticky notes that you can switch between.