Auto Debit Attempts is a company setting (Settings > Company > Billing/Payment) and a Client Group setting (Settings > Company > Client Groups). It appears to be ignored, and auto debit attempts are not limited. After the number of attempts has been reached without a successful charge, the Payment Account enabled for auto debit should be de-selected so that it will not attempt any further charges.
This setting is stored with a key of autodebit_attempts within client_group_settings and company_settings.
To reproduce, create a payment account that will fail. If testing with Stripe Payments use a card that should be declined per https://stripe.com/docs/testing#declined-payments and set a number of attempts to 1. Allow the cron to run to process the charge, receive the decline, note that the payment account is still enabled.
I'm not sure where the number of attempts is tracked.