This is not necessarily true. To be fair, I thought the same way a few years back! I was convinced that once the Fed started to hike, the longer end of the curve (10s and 30s) would drift toward higher yields... but that's not what happened. Since the Fed started hiking in December 2015, US 10s have gone from 2.2% to 1.4% to... 2.2%.
What ended up happening was that the US curve flattened sharply; front-end yields drifted higher as the Fed hiked, but the back end stayed rangebound because bond traders were confident that the terminal rate (the rate when the Fed stops hiking) would be somewhere in the high-2s / low-3s.
If the US economy really starts to rally, and people move their expectations of the terminal rate higher, then we might see bond yields drop. But at the moment, I really don't think there's a lot of scope for (US) bond markets to collapse
(Japanese and European bonds are another matter.)