Are the breaking of bonds simultaneously endothermic and exothermic?
Bond breaking does NOT release energy!!

In the course of a chemical reaction, you typically break some bonds and form some new bonds. The bond breaking is always endothermic. The formation of new bonds is exothermic, so depending on whether the old bonds or the new bonds were stronger, the reaction overall can be either endothermic or exothermic.

When somebody says chemical bonds “contain potential energy” it is a HIGHLY misleading statement. You justifiably interpreted this as: when you break bonds, they release the potential energy that was stored within. This is, unfortunately, wrong. That phrase should honestly be banned, but somehow it has always persisted.

What that phrase really means is more nuanced. It means that some chemical bonds can be broken in a reaction which subsequently produces more energy via bond formation.

Bond breaking is always endothermic!

For further reading see: Is Bond Formation &quot;Strictly&quot; Exothermic?