[I_am_tiberius]

But there's no way the bitcoin network doesn't accept the settlement. A bank can just throw checks away if they like.

[bduerst]

Bitcoin will not accept a settlement if the wallet no longer has funds.

Banks throwing away checks is not a problem. Bouncing checks are a fraud problem, which is why most everyone have moved on from using checks is many situations.

Lightning's solution is to just 'watch' everyone you do transactions with, which is a lazy, non-viable solution analagous to continuously watching an anonymous stranger's bank account when they write you a check.