I worked on mainly ACH processes (a very necessary component, though not one that would have cause global thermonuclear war on Jan 1, 2000 or Sep 9, 1999) and many of them had the 2 year digits in ACH processing date fields. Plus there were back-end calculations made against dates stored in string variables (a problem with the 2-year date formats), though those were rare (mainly in database queries sorting records by date if it was stored as a string and processing records in the wrong sequence, perhaps halting operation). Most of what I worked on was string-to-date conversions to and from data files (mainly ACH files).
Does that sound like what you worked on?
Most were routine, but quite a few were life-threatening. While banks probably wouldn't lose your account balances, they could very well have frozen accounts, transferred wrong amounts, thinking there was an overdue amount, or transferred not enough thinking that it had already been done. The economy could have been ruined.