I'm so happy someone has raised these questions. I don't have any experience with such negotiations yet, but I do plan to make schools aware of my other funding offers with the expectation that I'll be offered more competitive packages if possible. I was lucky to be told by a professor who contacted me with an offer to think of him as my "agent" - he wanted me to know that the school was willing to be aggressively competitive with their offer, and that I should feel comfortable disclosing competitive offers so that he could use the info as a lever for more money for me from the dean. This came from someone at a state school, a UC, for what it's worth.
I don't know to what extent this varies from institution to institution, but I do think it's appropriate to be savvy in negotiating - just as you would any other job offer - and to be critical of these funding packages, which in many ways are exploitative. We are made to feel like gods for being offered a shot at functioning as cheap labor, and I think the competitiveness of being offered admission to these programs makes us feel meek about questioning some of their terms.