I know, because I have used one! Completely unbranded, both physically (on the case), and software wise (no branding of ANY sort in the OS).
How is this possible? Well, I'm a senior technical analyst for PC World in India (the computer magazine, from IDG,
www.idg.com), and we get evaluation units from RIM. They have a little sticker on the back that says "Property of Research in Motion. Evaluation only, not for sale or lease". We get them for a month or so, and then return them.
And they work perfectly with any SIM that you put in. If the SIM has the BB service setup by the carrier, it works fine, and if it doesn't, it works just like a normal phone. I know this for a fact, since I have personally used SIMs from Airtel, Vodafone, and Idea (all big in India), and they have worked perfectly.
I have upgraded the OS on these phones, and they continue to have no branding. The vendor ID is -1
So yes, such phones do exist.