in some of the WWII piston fighters like the P-47. However, it simply couldn't happen because the planes had too much drag. Their confusion was due to instrumentation errors that caused abnormally high indicated airspeed in the dives.

Several of those fighters did run into compressability problems due to their thick wings. The airflow on the upper surface would reach sonic speed in a high speed dive, causing shock waves. These shock waves might disrupt the airflow over the tail. Early P-38s were the first (that I know of) to encounter this problem. Some of the planes were destroyed. P-47s also had the problem.

As for the claim that the XF-86 broke Mach 1 before Yeager, improper instrumentation makes the claim impossible to prove. It might've happened but we may never know for sure.