Not knowing anything about your lease, they'd generally have to sue you for the fee. The trash fee goes to the owner of the property, which is your landlord presumably. That means they have to pay it and then try to get the money from you somehow. If they have proof, they can show it to the court and have the judge decide if you, in fact, do owe the fee because you were responsible; however, if they don't have proof, the burden of proof is on them and that makes it harder for the court to rule against you.
Now, if they take you to court and win, you may have to pay the court and legal fees as well as the penalty, but if they lose, they'll have to absorb the cost.
Furthermore, if your lease says anything to the contrary, the lease will override whatever the general rule is. Unfortunately, the fact that your neighbors leave trash out doesn't help your case unless it does go to court, in which you're going to try to convince the judge that it was more likely than not the trash of your neighbor, not you.
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