They seem to get one of two messages: “You can’t add any more of _____ to your cart” or “There are no more _____ available to purchase”.
Generally, we have anywhere from 1 to 5 in stock of these products, this time around we had 4 of them. They are all restricted to 1 per customer via Woocommerce Purchase Limits (1.2).
We run on Divi theme (latest update) and we have a few Woocommerce related plugins: Woocommerce Menu Cart (2.7.5), Woocommerce Extended Coupons PRO (2.6.1.1), Woocommerce Purchase Limits (1.2), Woocommerce Authorize.net AIM Premium (9.6), and NextMove Lite (2.9.6).