Can I fail a partially paid invoice?

It is not possible to fail a partially-paid invoice. Two alternatives are available for closing these invoices:

1) Post another "External Payment" against the invoice to close out the remaining balance. This method offers a dropdown and note field where you can indicate it's a bad debt or write-off. Once the remaining value is accounted for, the invoice will close as paid.

2) Delete or void the partial payment, and then mark the invoice as failed. To delete a partial payment, view the invoice and click on the timestamp beside the payment (below the invoice) to view that transaction's details. You can void or delete the transaction from the top of that transaction details page.