The short version: Mt. Gox stopped paying out USD-denominated claims, because (as was correctly perceived by many people) they were insolvent. They continued paying out BTC-denominated claims. The only way to get value out of Mt. Gox was to either go through yen (which the vast majority of customers couldn't do [0]) or buy Bitcoin and withdraw, which caused the price of Bitcoin to gallop upwards in late 2013.
[0] I got a number of interesting business propositions that year: https://twitter.com/patio11/status/423869016776933376
Didn't the price crash after the extreme rise? Or was this because the BTC was stolen and unrelated to the USD insolvency? Was the "good for BTC" tongue in cheek?
Similar thing could happen with people redeeming USDT for USD if they don't have enough to cover the already issued USDT.