Sure, you can try to negotiate pretty much anything. But what kind of leverage do you think you have? Insurance companies can put many millions of dollars of revenue at risk with the threat of dropping a stubborn healthcare provider from their network.
In contrast, your negotiation position is A) please? and B) I may be forced to declare bankruptcy, in which case you may or may not get your money. If you have enough assets, then option B is basically another form of A: "please, don't take away my house or the savings I've spent my life earning." The bigger your bill, the greater the incentive to deal with the annoyance involved in collecting from you. The likely consequence of your personal financial ruin doesn't enter into their calculations.