How do you know who has rental insurance and who does not??
Obviously, since the fire was your fault, you are responsible for the damages.
Is it the apartment owners that want you to pay the damages or the renters below you or both?
If this is a large sum of money, you may want to speak to a lawyer about your responsibilities to see if you can be responsible for others deductibles on their policies only and not the full amount of damage costs.
As an added note...to all renters reading this, renters insurance is really quite reasonable, compared to this. Check it out. Call your auto insurance co. about renters insurance.
I really don't know how you can get out of paying for damage caused by a fire in your apartment.
As for the landlord having insurance...probably do, don't know if they have to file this claim or not with their insurance because when they do, their insurance premiums will most likely rise.
Even if they do file, there will be a deductible to pay, could be over $1000. You would be responsible for that too.
As I said, speak with a lawyer and find out how much you are responsible for. If you do nothing, they will take you to court and I expect you will lose.