Twice now I've encountered burned power connectors on older ATX motherboards. The first was a pair of Tyan S2460 boards with dual Athlon MP CPUs. The owner was having problems with them and thought one of the CPUs had failed. He gave them to me and while testing I noticed the problem. Both of these boards were designed to the original ATX specifications and only had a 20-pin power connector. The CPU regulators used +5 volts as their input supply to generate the CPU voltages. The increasing power requirements of CPUs in this era (especially dual CPUs) resulted in increased current (amperes) through the connectors. Since increasing current results in increasing heat dissipation (due to wire and connector resistance) and the increased heat also increases the resistance the same, a thermal runaway condition can occur. This greatly reduces the life of them - the plastic of the connectors bakes and melts and eventually the solder on the motherboard connector pins melt. All of this is bad for stability.
I deemed the boards were useless and, not finding any economical Socket A replacements, decided to upgrade two other systems and give the rest of the parts to a PC recycling center. One of the CPUs, an XP 1800, replaced a 1.3GHz Thunderbird in a Gigabyte GA-7ZXE. This solved a problem with it's Nvidia 6600GT AGP video card and the newer drivers requiring SSE support. Everything was good for a few months but eventually instability set in. I found the +5V pins (red wires) on the ATX connectors had burned and fused together. The 6600GT had it's own power connector and I didn't think one CPU would cause an overload on the ATX connector. Burned twice you might say.
Newer versions of the ATX specification, ATX12V, added the second four-pin CPU power connector that uses +12V instead which greatly reduces the problem. In electrical physics, for the same amount of power (P, in watts), doubling the voltage (V) halves the current (I). The voltage (the amount of energy each electron is carrying) doesn't affect wire or connector heating, only the current (the quantity of electrons moving through the wire) and the conductor's resistance (R, in ohms) does. This heating effect is described by the formula P=I²R and is called "I squared R" losses. Something to keep in mind when upgrading or recycling old systems.
Note that this problem doesn't just affect the connectors. I almost fried a 400W ATX power supply while testing the Tyans with two CPUs because of the current requirements on the +5V rail was more than it could handle.