How do you negotiate your severance? I don't see the personbeing terminated as having much leverage. Are companies really that afraid that you'll bad mouth them? I'm struggling to imagine that conversation:
"Here is 8 weeks compensation, in return for signing this non-disparagement agreement"
"No, I want 12 weeks compensation or I won't sign your agreement"
Yup, basically right. It is still a legal document too. Typically severance documents will have a deadline to be signed which is after your official termination date. You're fully entitled to not sign anything at your termination so you can consult with counsel. If the stakes are high enough, a proper employment lawyer can advise you on proper next steps given your severance documents (what you're entitled to for their asks (non-competes? confidentiality? agreeing to company-run arbitration in any future matters?, etc).
But, once you counter, they have the option to re-consider their entire offer, including perhaps withdrawing it completely if they consider the transaction no longer worth it. Then you're still dismissed and get nothing.
All of this is assuming policies under american at-will employment. Any other contracts or government regulations can alter matters. Some countries even have mandatory severance of a year or more when fired, but those countries tend to not have great economic growth prospects (that's where Mitt Romney once said "I like being able to fire people" with the implication of otherwise dead weight just sits around knowing it can do nothing and still get paid).
"Here is 8 weeks compensation, in return for signing this non-disparagement agreement"
"No, I want 12 weeks compensation or I won't sign your agreement"