Only if you buy a shitty PSU. Quality ones have lots of protection circuits so worst case the PC would randomly shut down in high load situations.
Even then if the manufacturer put two connectors on the end of the cable they made sure to put large enough conductors to handle at least the max load of the connectors specifications. Again if it’s not a shitty no name brand. Probably with a good safety factor as well.
It’s not like 300-400W is that much energy in the grand scheme of things, so thicker wires wouldn’t even be expensive.